# 线性代数

julia> A = [1 2 3; 4 1 6; 7 8 1]
3×3 Array{Int64,2}:
1  2  3
4  1  6
7  8  1

julia> tr(A)
3

julia> det(A)
104.0

julia> inv(A)
3×3 Array{Float64,2}:
-0.451923   0.211538    0.0865385
0.365385  -0.192308    0.0576923
0.240385   0.0576923  -0.0673077

julia> A = [-4. -17.; 2. 2.]
2×2 Array{Float64,2}:
-4.0  -17.0
2.0    2.0

julia> eigvals(A)
2-element Array{Complex{Float64},1}:
-1.0 + 5.0im
-1.0 - 5.0im

julia> eigvecs(A)
2×2 Array{Complex{Float64},2}:
0.945905+0.0im        0.945905-0.0im
-0.166924-0.278207im  -0.166924+0.278207im

julia> A = [1.5 2 -4; 3 -1 -6; -10 2.3 4]
3×3 Array{Float64,2}:
1.5   2.0  -4.0
3.0  -1.0  -6.0
-10.0   2.3   4.0

julia> factorize(A)
LU{Float64,Array{Float64,2}}
L factor:
3×3 Array{Float64,2}:
1.0    0.0       0.0
-0.15   1.0       0.0
-0.3   -0.132196  1.0
U factor:
3×3 Array{Float64,2}:
-10.0  2.3     4.0
0.0  2.345  -3.4
0.0  0.0    -5.24947

julia> B = [1.5 2 -4; 2 -1 -3; -4 -3 5]
3×3 Array{Float64,2}:
1.5   2.0  -4.0
2.0  -1.0  -3.0
-4.0  -3.0   5.0

julia> factorize(B)
BunchKaufman{Float64,Array{Float64,2}}
D factor:
3×3 Tridiagonal{Float64,Array{Float64,1}}:
-1.64286   0.0   ⋅
0.0      -2.8  0.0
⋅        0.0  5.0
U factor:
3×3 UnitUpperTriangular{Float64,Array{Float64,2}}:
1.0  0.142857  -0.8
⋅   1.0       -0.6
⋅    ⋅         1.0
permutation:
3-element Array{Int64,1}:
1
2
3

julia> B = [1.5 2 -4; 2 -1 -3; -4 -3 5]
3×3 Array{Float64,2}:
1.5   2.0  -4.0
2.0  -1.0  -3.0
-4.0  -3.0   5.0

julia> sB = Symmetric(B)
3×3 Symmetric{Float64,Array{Float64,2}}:
1.5   2.0  -4.0
2.0  -1.0  -3.0
-4.0  -3.0   5.0

sB 已经被标记成（实）对称矩阵，所以对于之后可能在它上面执行的操作，例如特征因子化或矩阵-向量乘积，只引用矩阵的一半可以提高效率。举个例子：

julia> B = [1.5 2 -4; 2 -1 -3; -4 -3 5]
3×3 Array{Float64,2}:
1.5   2.0  -4.0
2.0  -1.0  -3.0
-4.0  -3.0   5.0

julia> sB = Symmetric(B)
3×3 Symmetric{Float64,Array{Float64,2}}:
1.5   2.0  -4.0
2.0  -1.0  -3.0
-4.0  -3.0   5.0

julia> x = [1; 2; 3]
3-element Array{Int64,1}:
1
2
3

julia> sB\x
3-element Array{Float64,1}:
-1.7391304347826084
-1.1086956521739126
-1.4565217391304346

\ 运算在这里执行线性求解。左除运算符相当强大，很容易写出紧凑、可读的代码，它足够灵活，可以求解各种线性方程组。

## 特殊矩阵

Symmetric对称矩阵
Hermitian埃尔米特矩阵
UpperTriangular三角矩阵
LowerTriangular三角矩阵
Tridiagonal三对角矩阵
SymTridiagonal对称三对角矩阵
Bidiagonal上/下双对角矩阵
Diagonal对角矩阵
UniformScaling等比缩放运算符

### 基本运算

SymmetricMVinv, sqrt, exp
HermitianMVinv, sqrt, exp
UpperTriangularMVMVinv, det
LowerTriangularMVMVinv, det
SymTridiagonalMMMSMVeigmax, eigmin
TridiagonalMMMSMV
BidiagonalMMMSMV
DiagonalMMMVMVinv, det, logdet, /
UniformScalingMMMVSMVS/

M (matrix)针对矩阵与矩阵运算的优化方法可用
V (vector)针对矩阵与向量运算的优化方法可用
S (scalar)针对矩阵与标量运算的优化方法可用

### 矩阵分解

SymmetricSYARI
HermitianHEARI
UpperTriangularTRAAA
LowerTriangularTRAAA
SymTridiagonalSTAARIAV
TridiagonalGT
BidiagonalBDAA
DiagonalDIA

A (all)找到所有特征值和/或特征向量的优化方法可用例如，eigvals(M)
R (range)通过第 ih 个特征值寻找第 il 个特征值的优化方法可用eigvals(M, il, ih)
I (interval)寻找在区间 [vl, vh] 内的特征值的优化方法可用eigvals(M, vl, vh)
V (vectors)寻找对应于特征值 x=[x1, x2,...] 的特征向量的优化方法可用eigvecs(M, x)

### 等比缩放运算符

UniformScaling 运算符代表一个标量乘以恒同运算符，λ*I。恒同运算符 I 定义为常量和 UniformScaling 的实例。这些运算符是通用的，并且会在二元运算符 +-*\ 中与另一个矩阵相匹配。对于 A+IA-I ，这意味着 A 必须是个方阵。与恒同运算符 I 相乘是一个空操作（除了检查比例因子是一），因此几乎没有开销。

julia> U = UniformScaling(2);

julia> a = [1 2; 3 4]
2×2 Array{Int64,2}:
1  2
3  4

julia> a + U
2×2 Array{Int64,2}:
3  2
3  6

julia> a * U
2×2 Array{Int64,2}:
2  4
6  8

julia> [a U]
2×4 Array{Int64,2}:
1  2  2  0
3  4  0  2

julia> b = [1 2 3; 4 5 6]
2×3 Array{Int64,2}:
1  2  3
4  5  6

julia> b - U
ERROR: DimensionMismatch("matrix is not square: dimensions are (2, 3)")
Stacktrace:
[...]

## 矩阵分解

CholeskyCholesky 分解
CholeskyPivotedPivoted Cholesky 分解
LULU 分解
LUTridiagonal针对 Tridiagonal 矩阵的 LU 分解
QRQR 分解
QRCompactWYQR 分解的紧凑 WY 形式
QRPivotedPivoted QR 分解
HessenbergHessenberg 分解
Eigen谱分解
SVD奇异值分解
GeneralizedSVD广义 SVD

## 标准函数

Julia 中的线性代数函数主要通过调用 LAPACK 中的函数来实现。稀疏分解则调用 SuiteSparse 中的函数。

Base.:*Method
*(A::AbstractMatrix, B::AbstractMatrix)

Matrix multiplication.

Examples

julia> [1 1; 0 1] * [1 0; 1 1]
2×2 Array{Int64,2}:
2  1
1  1
Base.:\Method
\(A, B)

Matrix division using a polyalgorithm. For input matrices A and B, the result X is such that A*X == B when A is square. The solver that is used depends upon the structure of A. If A is upper or lower triangular (or diagonal), no factorization of A is required and the system is solved with either forward or backward substitution. For non-triangular square matrices, an LU factorization is used.

For rectangular A the result is the minimum-norm least squares solution computed by a pivoted QR factorization of A and a rank estimate of A based on the R factor.

When A is sparse, a similar polyalgorithm is used. For indefinite matrices, the LDLt factorization does not use pivoting during the numerical factorization and therefore the procedure can fail even for invertible matrices.

Examples

julia> A = [1 0; 1 -2]; B = [32; -4];

julia> X = A \ B
2-element Array{Float64,1}:
32.0
18.0

julia> A * X == B
true
LinearAlgebra.dotFunction
dot(x, y)
x ⋅ y

Compute the dot product between two vectors. For complex vectors, the first vector is conjugated.

dot also works on arbitrary iterable objects, including arrays of any dimension, as long as dot is defined on the elements.

dot is semantically equivalent to sum(dot(vx,vy) for (vx,vy) in zip(x, y)), with the added restriction that the arguments must have equal lengths.

x ⋅ y (where ⋅ can be typed by tab-completing \cdot in the REPL) is a synonym for dot(x, y).

Examples

julia> dot([1; 1], [2; 3])
5

julia> dot([im; im], [1; 1])
0 - 2im

julia> dot(1:5, 2:6)
70

julia> x = fill(2., (5,5));

julia> y = fill(3., (5,5));

julia> dot(x, y)
150.0
LinearAlgebra.crossFunction
cross(x, y)
×(x,y)

Compute the cross product of two 3-vectors.

Examples

julia> a = [0;1;0]
3-element Array{Int64,1}:
0
1
0

julia> b = [0;0;1]
3-element Array{Int64,1}:
0
0
1

julia> cross(a,b)
3-element Array{Int64,1}:
1
0
0
LinearAlgebra.factorizeFunction
factorize(A)

Compute a convenient factorization of A, based upon the type of the input matrix. factorize checks A to see if it is symmetric/triangular/etc. if A is passed as a generic matrix. factorize checks every element of A to verify/rule out each property. It will short-circuit as soon as it can rule out symmetry/triangular structure. The return value can be reused for efficient solving of multiple systems. For example: A=factorize(A); x=A\b; y=A\C.

Properties of Atype of factorization
Positive-definiteCholesky (see cholesky)
Dense Symmetric/HermitianBunch-Kaufman (see bunchkaufman)
Sparse Symmetric/HermitianLDLt (see ldlt)
TriangularTriangular
DiagonalDiagonal
BidiagonalBidiagonal
TridiagonalLU (see lu)
Symmetric real tridiagonalLDLt (see ldlt)
General squareLU (see lu)
General non-squareQR (see qr)

If factorize is called on a Hermitian positive-definite matrix, for instance, then factorize will return a Cholesky factorization.

Examples

julia> A = Array(Bidiagonal(fill(1.0, (5, 5)), :U))
5×5 Array{Float64,2}:
1.0  1.0  0.0  0.0  0.0
0.0  1.0  1.0  0.0  0.0
0.0  0.0  1.0  1.0  0.0
0.0  0.0  0.0  1.0  1.0
0.0  0.0  0.0  0.0  1.0

julia> factorize(A) # factorize will check to see that A is already factorized
5×5 Bidiagonal{Float64,Array{Float64,1}}:
1.0  1.0   ⋅    ⋅    ⋅
⋅   1.0  1.0   ⋅    ⋅
⋅    ⋅   1.0  1.0   ⋅
⋅    ⋅    ⋅   1.0  1.0
⋅    ⋅    ⋅    ⋅   1.0

This returns a 5×5 Bidiagonal{Float64}, which can now be passed to other linear algebra functions (e.g. eigensolvers) which will use specialized methods for Bidiagonal types.

LinearAlgebra.DiagonalType
Diagonal(A::AbstractMatrix)

Construct a matrix from the diagonal of A.

Examples

julia> A = [1 2 3; 4 5 6; 7 8 9]
3×3 Array{Int64,2}:
1  2  3
4  5  6
7  8  9

julia> Diagonal(A)
3×3 Diagonal{Int64,Array{Int64,1}}:
1  ⋅  ⋅
⋅  5  ⋅
⋅  ⋅  9
Diagonal(V::AbstractVector)

Construct a matrix with V as its diagonal.

Examples

julia> V = [1, 2]
2-element Array{Int64,1}:
1
2

julia> Diagonal(V)
2×2 Diagonal{Int64,Array{Int64,1}}:
1  ⋅
⋅  2
LinearAlgebra.BidiagonalType
Bidiagonal(dv::V, ev::V, uplo::Symbol) where V <: AbstractVector

Constructs an upper (uplo=:U) or lower (uplo=:L) bidiagonal matrix using the given diagonal (dv) and off-diagonal (ev) vectors. The result is of type Bidiagonal and provides efficient specialized linear solvers, but may be converted into a regular matrix with convert(Array, _) (or Array(_) for short). The length of ev must be one less than the length of dv.

Examples

julia> dv = [1, 2, 3, 4]
4-element Array{Int64,1}:
1
2
3
4

julia> ev = [7, 8, 9]
3-element Array{Int64,1}:
7
8
9

julia> Bu = Bidiagonal(dv, ev, :U) # ev is on the first superdiagonal
4×4 Bidiagonal{Int64,Array{Int64,1}}:
1  7  ⋅  ⋅
⋅  2  8  ⋅
⋅  ⋅  3  9
⋅  ⋅  ⋅  4

julia> Bl = Bidiagonal(dv, ev, :L) # ev is on the first subdiagonal
4×4 Bidiagonal{Int64,Array{Int64,1}}:
1  ⋅  ⋅  ⋅
7  2  ⋅  ⋅
⋅  8  3  ⋅
⋅  ⋅  9  4
Bidiagonal(A, uplo::Symbol)

Construct a Bidiagonal matrix from the main diagonal of A and its first super- (if uplo=:U) or sub-diagonal (if uplo=:L).

Examples

julia> A = [1 1 1 1; 2 2 2 2; 3 3 3 3; 4 4 4 4]
4×4 Array{Int64,2}:
1  1  1  1
2  2  2  2
3  3  3  3
4  4  4  4

julia> Bidiagonal(A, :U) # contains the main diagonal and first superdiagonal of A
4×4 Bidiagonal{Int64,Array{Int64,1}}:
1  1  ⋅  ⋅
⋅  2  2  ⋅
⋅  ⋅  3  3
⋅  ⋅  ⋅  4

julia> Bidiagonal(A, :L) # contains the main diagonal and first subdiagonal of A
4×4 Bidiagonal{Int64,Array{Int64,1}}:
1  ⋅  ⋅  ⋅
2  2  ⋅  ⋅
⋅  3  3  ⋅
⋅  ⋅  4  4
LinearAlgebra.SymTridiagonalType
SymTridiagonal(dv::V, ev::V) where V <: AbstractVector

Construct a symmetric tridiagonal matrix from the diagonal (dv) and first sub/super-diagonal (ev), respectively. The result is of type SymTridiagonal and provides efficient specialized eigensolvers, but may be converted into a regular matrix with convert(Array, _) (or Array(_) for short).

Examples

julia> dv = [1, 2, 3, 4]
4-element Array{Int64,1}:
1
2
3
4

julia> ev = [7, 8, 9]
3-element Array{Int64,1}:
7
8
9

julia> SymTridiagonal(dv, ev)
4×4 SymTridiagonal{Int64,Array{Int64,1}}:
1  7  ⋅  ⋅
7  2  8  ⋅
⋅  8  3  9
⋅  ⋅  9  4
SymTridiagonal(A::AbstractMatrix)

Construct a symmetric tridiagonal matrix from the diagonal and first sub/super-diagonal, of the symmetric matrix A.

Examples

julia> A = [1 2 3; 2 4 5; 3 5 6]
3×3 Array{Int64,2}:
1  2  3
2  4  5
3  5  6

julia> SymTridiagonal(A)
3×3 SymTridiagonal{Int64,Array{Int64,1}}:
1  2  ⋅
2  4  5
⋅  5  6
LinearAlgebra.TridiagonalType
Tridiagonal(dl::V, d::V, du::V) where V <: AbstractVector

Construct a tridiagonal matrix from the first subdiagonal, diagonal, and first superdiagonal, respectively. The result is of type Tridiagonal and provides efficient specialized linear solvers, but may be converted into a regular matrix with convert(Array, _) (or Array(_) for short). The lengths of dl and du must be one less than the length of d.

Examples

julia> dl = [1, 2, 3];

julia> du = [4, 5, 6];

julia> d = [7, 8, 9, 0];

julia> Tridiagonal(dl, d, du)
4×4 Tridiagonal{Int64,Array{Int64,1}}:
7  4  ⋅  ⋅
1  8  5  ⋅
⋅  2  9  6
⋅  ⋅  3  0
Tridiagonal(A)

Construct a tridiagonal matrix from the first sub-diagonal, diagonal and first super-diagonal of the matrix A.

Examples

julia> A = [1 2 3 4; 1 2 3 4; 1 2 3 4; 1 2 3 4]
4×4 Array{Int64,2}:
1  2  3  4
1  2  3  4
1  2  3  4
1  2  3  4

julia> Tridiagonal(A)
4×4 Tridiagonal{Int64,Array{Int64,1}}:
1  2  ⋅  ⋅
1  2  3  ⋅
⋅  2  3  4
⋅  ⋅  3  4
LinearAlgebra.SymmetricType
Symmetric(A, uplo=:U)

Construct a Symmetric view of the upper (if uplo = :U) or lower (if uplo = :L) triangle of the matrix A.

Examples

julia> A = [1 0 2 0 3; 0 4 0 5 0; 6 0 7 0 8; 0 9 0 1 0; 2 0 3 0 4]
5×5 Array{Int64,2}:
1  0  2  0  3
0  4  0  5  0
6  0  7  0  8
0  9  0  1  0
2  0  3  0  4

julia> Supper = Symmetric(A)
5×5 Symmetric{Int64,Array{Int64,2}}:
1  0  2  0  3
0  4  0  5  0
2  0  7  0  8
0  5  0  1  0
3  0  8  0  4

julia> Slower = Symmetric(A, :L)
5×5 Symmetric{Int64,Array{Int64,2}}:
1  0  6  0  2
0  4  0  9  0
6  0  7  0  3
0  9  0  1  0
2  0  3  0  4

Note that Supper will not be equal to Slower unless A is itself symmetric (e.g. if A == transpose(A)).

LinearAlgebra.HermitianType
Hermitian(A, uplo=:U)

Construct a Hermitian view of the upper (if uplo = :U) or lower (if uplo = :L) triangle of the matrix A.

Examples

julia> A = [1 0 2+2im 0 3-3im; 0 4 0 5 0; 6-6im 0 7 0 8+8im; 0 9 0 1 0; 2+2im 0 3-3im 0 4];

julia> Hupper = Hermitian(A)
5×5 Hermitian{Complex{Int64},Array{Complex{Int64},2}}:
1+0im  0+0im  2+2im  0+0im  3-3im
0+0im  4+0im  0+0im  5+0im  0+0im
2-2im  0+0im  7+0im  0+0im  8+8im
0+0im  5+0im  0+0im  1+0im  0+0im
3+3im  0+0im  8-8im  0+0im  4+0im

julia> Hlower = Hermitian(A, :L)
5×5 Hermitian{Complex{Int64},Array{Complex{Int64},2}}:
1+0im  0+0im  6+6im  0+0im  2-2im
0+0im  4+0im  0+0im  9+0im  0+0im
6-6im  0+0im  7+0im  0+0im  3+3im
0+0im  9+0im  0+0im  1+0im  0+0im
2+2im  0+0im  3-3im  0+0im  4+0im

Note that Hupper will not be equal to Hlower unless A is itself Hermitian (e.g. if A == adjoint(A)).

All non-real parts of the diagonal will be ignored.

Hermitian(fill(complex(1,1), 1, 1)) == fill(1, 1, 1)
LinearAlgebra.LowerTriangularType
LowerTriangular(A::AbstractMatrix)

Construct a LowerTriangular view of the matrix A.

Examples

julia> A = [1.0 2.0 3.0; 4.0 5.0 6.0; 7.0 8.0 9.0]
3×3 Array{Float64,2}:
1.0  2.0  3.0
4.0  5.0  6.0
7.0  8.0  9.0

julia> LowerTriangular(A)
3×3 LowerTriangular{Float64,Array{Float64,2}}:
1.0   ⋅    ⋅
4.0  5.0   ⋅
7.0  8.0  9.0
LinearAlgebra.UpperTriangularType
UpperTriangular(A::AbstractMatrix)

Construct an UpperTriangular view of the matrix A.

Examples

julia> A = [1.0 2.0 3.0; 4.0 5.0 6.0; 7.0 8.0 9.0]
3×3 Array{Float64,2}:
1.0  2.0  3.0
4.0  5.0  6.0
7.0  8.0  9.0

julia> UpperTriangular(A)
3×3 UpperTriangular{Float64,Array{Float64,2}}:
1.0  2.0  3.0
⋅   5.0  6.0
⋅    ⋅   9.0
LinearAlgebra.UniformScalingType
UniformScaling{T<:Number}

Generically sized uniform scaling operator defined as a scalar times the identity operator, λ*I. See also I.

Examples

julia> J = UniformScaling(2.)
UniformScaling{Float64}
2.0*I

julia> A = [1. 2.; 3. 4.]
2×2 Array{Float64,2}:
1.0  2.0
3.0  4.0

julia> J*A
2×2 Array{Float64,2}:
2.0  4.0
6.0  8.0
LinearAlgebra.luFunction
lu(A, pivot=Val(true); check = true) -> F::LU

Compute the LU factorization of A.

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition's validity (via issuccess) lies with the user.

In most cases, if A is a subtype S of AbstractMatrix{T} with an element type T supporting +, -, * and /, the return type is LU{T,S{T}}. If pivoting is chosen (default) the element type should also support abs and <.

The individual components of the factorization F can be accessed via getproperty:

ComponentDescription
F.LL (lower triangular) part of LU
F.UU (upper triangular) part of LU
F.p(right) permutation Vector
F.P(right) permutation Matrix

Iterating the factorization produces the components F.L, F.U, and F.p.

The relationship between F and A is

F.L*F.U == A[F.p, :]

F further supports the following functions:

Supported functionLULU{T,Tridiagonal{T}}
/
\
inv
det
logdet
logabsdet
size

Examples

julia> A = [4 3; 6 3]
2×2 Array{Int64,2}:
4  3
6  3

julia> F = lu(A)
LU{Float64,Array{Float64,2}}
L factor:
2×2 Array{Float64,2}:
1.0       0.0
0.666667  1.0
U factor:
2×2 Array{Float64,2}:
6.0  3.0
0.0  1.0

julia> F.L * F.U == A[F.p, :]
true

julia> l, u, p = lu(A); # destructuring via iteration

julia> l == F.L && u == F.U && p == F.p
true
LinearAlgebra.lu!Function
lu!(A, pivot=Val(true); check = true) -> LU

lu! is the same as lu, but saves space by overwriting the input A, instead of creating a copy. An InexactError exception is thrown if the factorization produces a number not representable by the element type of A, e.g. for integer types.

Examples

julia> A = [4. 3.; 6. 3.]
2×2 Array{Float64,2}:
4.0  3.0
6.0  3.0

julia> F = lu!(A)
LU{Float64,Array{Float64,2}}
L factor:
2×2 Array{Float64,2}:
1.0       0.0
0.666667  1.0
U factor:
2×2 Array{Float64,2}:
6.0  3.0
0.0  1.0

julia> iA = [4 3; 6 3]
2×2 Array{Int64,2}:
4  3
6  3

julia> lu!(iA)
ERROR: InexactError: Int64(0.6666666666666666)
Stacktrace:
[...]
LinearAlgebra.choleskyFunction
cholesky(A, Val(false); check = true) -> Cholesky

Compute the Cholesky factorization of a dense symmetric positive definite matrix A and return a Cholesky factorization. The matrix A can either be a Symmetric or Hermitian StridedMatrix or a perfectly symmetric or Hermitian StridedMatrix. The triangular Cholesky factor can be obtained from the factorization F with: F.L and F.U. The following functions are available for Cholesky objects: size, \, inv, det, logdet and isposdef.

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition's validity (via issuccess) lies with the user.

Examples

julia> A = [4. 12. -16.; 12. 37. -43.; -16. -43. 98.]
3×3 Array{Float64,2}:
4.0   12.0  -16.0
12.0   37.0  -43.0
-16.0  -43.0   98.0

julia> C = cholesky(A)
Cholesky{Float64,Array{Float64,2}}
U factor:
3×3 UpperTriangular{Float64,Array{Float64,2}}:
2.0  6.0  -8.0
⋅   1.0   5.0
⋅    ⋅    3.0

julia> C.U
3×3 UpperTriangular{Float64,Array{Float64,2}}:
2.0  6.0  -8.0
⋅   1.0   5.0
⋅    ⋅    3.0

julia> C.L
3×3 LowerTriangular{Float64,Array{Float64,2}}:
2.0   ⋅    ⋅
6.0  1.0   ⋅
-8.0  5.0  3.0

julia> C.L * C.U == A
true
cholesky(A, Val(true); tol = 0.0, check = true) -> CholeskyPivoted

Compute the pivoted Cholesky factorization of a dense symmetric positive semi-definite matrix A and return a CholeskyPivoted factorization. The matrix A can either be a Symmetric or Hermitian StridedMatrix or a perfectly symmetric or Hermitian StridedMatrix. The triangular Cholesky factor can be obtained from the factorization F with: F.L and F.U. The following functions are available for CholeskyPivoted objects: size, \, inv, det, and rank. The argument tol determines the tolerance for determining the rank. For negative values, the tolerance is the machine precision.

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition's validity (via issuccess) lies with the user.

LinearAlgebra.cholesky!Function
cholesky!(A, Val(false); check = true) -> Cholesky

The same as cholesky, but saves space by overwriting the input A, instead of creating a copy. An InexactError exception is thrown if the factorization produces a number not representable by the element type of A, e.g. for integer types.

Examples

julia> A = [1 2; 2 50]
2×2 Array{Int64,2}:
1   2
2  50

julia> cholesky!(A)
ERROR: InexactError: Int64(6.782329983125268)
Stacktrace:
[...]
cholesky!(A, Val(true); tol = 0.0, check = true) -> CholeskyPivoted

The same as cholesky, but saves space by overwriting the input A, instead of creating a copy. An InexactError exception is thrown if the factorization produces a number not representable by the element type of A, e.g. for integer types.

LinearAlgebra.lowrankupdateFunction
lowrankupdate(C::Cholesky, v::StridedVector) -> CC::Cholesky

Update a Cholesky factorization C with the vector v. If A = C.U'C.U then CC = cholesky(C.U'C.U + v*v') but the computation of CC only uses O(n^2) operations.

LinearAlgebra.lowrankdowndateFunction
lowrankdowndate(C::Cholesky, v::StridedVector) -> CC::Cholesky

Downdate a Cholesky factorization C with the vector v. If A = C.U'C.U then CC = cholesky(C.U'C.U - v*v') but the computation of CC only uses O(n^2) operations.

LinearAlgebra.lowrankupdate!Function
lowrankupdate!(C::Cholesky, v::StridedVector) -> CC::Cholesky

Update a Cholesky factorization C with the vector v. If A = C.U'C.U then CC = cholesky(C.U'C.U + v*v') but the computation of CC only uses O(n^2) operations. The input factorization C is updated in place such that on exit C == CC. The vector v is destroyed during the computation.

LinearAlgebra.lowrankdowndate!Function
lowrankdowndate!(C::Cholesky, v::StridedVector) -> CC::Cholesky

Downdate a Cholesky factorization C with the vector v. If A = C.U'C.U then CC = cholesky(C.U'C.U - v*v') but the computation of CC only uses O(n^2) operations. The input factorization C is updated in place such that on exit C == CC. The vector v is destroyed during the computation.

LinearAlgebra.ldltFunction
ldlt(S::SymTridiagonal) -> LDLt

Compute an LDLt factorization of the real symmetric tridiagonal matrix S such that S = L*Diagonal(d)*L' where L is a unit lower triangular matrix and d is a vector. The main use of an LDLt factorization F = ldlt(S) is to solve the linear system of equations Sx = b with F\b.

Examples

julia> S = SymTridiagonal([3., 4., 5.], [1., 2.])
3×3 SymTridiagonal{Float64,Array{Float64,1}}:
3.0  1.0   ⋅
1.0  4.0  2.0
⋅   2.0  5.0

julia> ldltS = ldlt(S);

julia> b = [6., 7., 8.];

julia> ldltS \ b
3-element Array{Float64,1}:
1.7906976744186047
0.627906976744186
1.3488372093023255

julia> S \ b
3-element Array{Float64,1}:
1.7906976744186047
0.627906976744186
1.3488372093023255
LinearAlgebra.ldlt!Function
ldlt!(S::SymTridiagonal) -> LDLt

Same as ldlt, but saves space by overwriting the input S, instead of creating a copy.

Examples

julia> S = SymTridiagonal([3., 4., 5.], [1., 2.])
3×3 SymTridiagonal{Float64,Array{Float64,1}}:
3.0  1.0   ⋅
1.0  4.0  2.0
⋅   2.0  5.0

julia> ldltS = ldlt!(S);

julia> ldltS === S
false

julia> S
3×3 SymTridiagonal{Float64,Array{Float64,1}}:
3.0       0.333333   ⋅
0.333333  3.66667   0.545455
⋅        0.545455  3.90909
LinearAlgebra.qrFunction
qr(A, pivot=Val(false)) -> F

Compute the QR factorization of the matrix A: an orthogonal (or unitary if A is complex-valued) matrix Q, and an upper triangular matrix R such that

$A = Q R$

The returned object F stores the factorization in a packed format:

The individual components of the decomposition F can be retrieved via property accessors:

• F.Q: the orthogonal/unitary matrix Q
• F.R: the upper triangular matrix R
• F.p: the permutation vector of the pivot (QRPivoted only)
• F.P: the permutation matrix of the pivot (QRPivoted only)

Iterating the decomposition produces the components Q, R, and if extant p.

The following functions are available for the QR objects: inv, size, and \. When A is rectangular, \ will return a least squares solution and if the solution is not unique, the one with smallest norm is returned. When A is not full rank, factorization with (column) pivoting is required to obtain a minimum norm solution.

Multiplication with respect to either full/square or non-full/square Q is allowed, i.e. both F.Q*F.R and F.Q*A are supported. A Q matrix can be converted into a regular matrix with Matrix. This operation returns the "thin" Q factor, i.e., if A is m×n with m>=n, then Matrix(F.Q) yields an m×n matrix with orthonormal columns. To retrieve the "full" Q factor, an m×m orthogonal matrix, use F.Q*Matrix(I,m,m). If m<=n, then Matrix(F.Q) yields an m×m orthogonal matrix.

Examples

julia> A = [3.0 -6.0; 4.0 -8.0; 0.0 1.0]
3×2 Array{Float64,2}:
3.0  -6.0
4.0  -8.0
0.0   1.0

julia> F = qr(A)
LinearAlgebra.QRCompactWY{Float64,Array{Float64,2}}
Q factor:
3×3 LinearAlgebra.QRCompactWYQ{Float64,Array{Float64,2}}:
-0.6   0.0   0.8
-0.8   0.0  -0.6
0.0  -1.0   0.0
R factor:
2×2 Array{Float64,2}:
-5.0  10.0
0.0  -1.0

julia> F.Q * F.R == A
true
Note

qr returns multiple types because LAPACK uses several representations that minimize the memory storage requirements of products of Householder elementary reflectors, so that the Q and R matrices can be stored compactly rather as two separate dense matrices.

LinearAlgebra.qr!Function
qr!(A, pivot=Val(false))

qr! is the same as qr when A is a subtype of StridedMatrix, but saves space by overwriting the input A, instead of creating a copy. An InexactError exception is thrown if the factorization produces a number not representable by the element type of A, e.g. for integer types.

Examples

julia> a = [1. 2.; 3. 4.]
2×2 Array{Float64,2}:
1.0  2.0
3.0  4.0

julia> qr!(a)
LinearAlgebra.QRCompactWY{Float64,Array{Float64,2}}
Q factor:
2×2 LinearAlgebra.QRCompactWYQ{Float64,Array{Float64,2}}:
-0.316228  -0.948683
-0.948683   0.316228
R factor:
2×2 Array{Float64,2}:
-3.16228  -4.42719
0.0      -0.632456

julia> a = [1 2; 3 4]
2×2 Array{Int64,2}:
1  2
3  4

julia> qr!(a)
ERROR: InexactError: Int64(-3.1622776601683795)
Stacktrace:
[...]
LinearAlgebra.QRType
QR <: Factorization

A QR matrix factorization stored in a packed format, typically obtained from qr. If $A$ is an m×n matrix, then

$A = Q R$

where $Q$ is an orthogonal/unitary matrix and $R$ is upper triangular. The matrix $Q$ is stored as a sequence of Householder reflectors $v_i$ and coefficients $\tau_i$ where:

$Q = \prod_{i=1}^{\min(m,n)} (I - \tau_i v_i v_i^T).$

Iterating the decomposition produces the components Q and R.

The object has two fields:

• factors is an m×n matrix.

• The upper triangular part contains the elements of $R$, that is R = triu(F.factors) for a QR object F.

• The subdiagonal part contains the reflectors $v_i$ stored in a packed format where $v_i$ is the $i$th column of the matrix V = I + tril(F.factors, -1).

• τ is a vector of length min(m,n) containing the coefficients $au_i$.

LinearAlgebra.QRCompactWYType
QRCompactWY <: Factorization

A QR matrix factorization stored in a compact blocked format, typically obtained from qr. If $A$ is an m×n matrix, then

$A = Q R$

where $Q$ is an orthogonal/unitary matrix and $R$ is upper triangular. It is similar to the QR format except that the orthogonal/unitary matrix $Q$ is stored in Compact WY format [Schreiber1989], as a lower trapezoidal matrix $V$ and an upper triangular matrix $T$ where

$Q = \prod_{i=1}^{\min(m,n)} (I - \tau_i v_i v_i^T) = I - V T V^T$

such that $v_i$ is the $i$th column of $V$, and $au_i$ is the $i$th diagonal element of $T$.

Iterating the decomposition produces the components Q and R.

The object has two fields:

• factors, as in the QR type, is an m×n matrix.

• The upper triangular part contains the elements of $R$, that is R = triu(F.factors) for a QR object F.

• The subdiagonal part contains the reflectors $v_i$ stored in a packed format such that V = I + tril(F.factors, -1).

• T is a square matrix with min(m,n) columns, whose upper triangular part gives the matrix $T$ above (the subdiagonal elements are ignored).

Note

This format should not to be confused with the older WY representation [Bischof1987].

LinearAlgebra.QRPivotedType
QRPivoted <: Factorization

A QR matrix factorization with column pivoting in a packed format, typically obtained from qr. If $A$ is an m×n matrix, then

$A P = Q R$

where $P$ is a permutation matrix, $Q$ is an orthogonal/unitary matrix and $R$ is upper triangular. The matrix $Q$ is stored as a sequence of Householder reflectors:

$Q = \prod_{i=1}^{\min(m,n)} (I - \tau_i v_i v_i^T).$

Iterating the decomposition produces the components Q, R, and p.

The object has three fields:

• factors is an m×n matrix.

• The upper triangular part contains the elements of $R$, that is R = triu(F.factors) for a QR object F.

• The subdiagonal part contains the reflectors $v_i$ stored in a packed format where $v_i$ is the $i$th column of the matrix V = I + tril(F.factors, -1).

• τ is a vector of length min(m,n) containing the coefficients $au_i$.

• jpvt is an integer vector of length n corresponding to the permutation $P$.

LinearAlgebra.lqFunction
lq(A) -> S::LQ

Compute the LQ decomposition of A. The decomposition's lower triangular component can be obtained from the LQ object S via S.L, and the orthogonal/unitary component via S.Q, such that A ≈ S.L*S.Q.

Iterating the decomposition produces the components S.L and S.Q.

The LQ decomposition is the QR decomposition of transpose(A).

Examples

julia> A = [5. 7.; -2. -4.]
2×2 Array{Float64,2}:
5.0   7.0
-2.0  -4.0

julia> S = lq(A)
LQ{Float64,Array{Float64,2}} with factors L and Q:
[-8.60233 0.0; 4.41741 -0.697486]
[-0.581238 -0.813733; -0.813733 0.581238]

julia> S.L * S.Q
2×2 Array{Float64,2}:
5.0   7.0
-2.0  -4.0

julia> l, q = S; # destructuring via iteration

julia> l == S.L &&  q == S.Q
true
LinearAlgebra.bunchkaufmanFunction
bunchkaufman(A, rook::Bool=false; check = true) -> S::BunchKaufman

Compute the Bunch-Kaufman [Bunch1977] factorization of a symmetric or Hermitian matrix A as P'*U*D*U'*P or P'*L*D*L'*P, depending on which triangle is stored in A, and return a BunchKaufman object. Note that if A is complex symmetric then U' and L' denote the unconjugated transposes, i.e. transpose(U) and transpose(L).

Iterating the decomposition produces the components S.D, S.U or S.L as appropriate given S.uplo, and S.p.

If rook is true, rook pivoting is used. If rook is false, rook pivoting is not used.

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition's validity (via issuccess) lies with the user.

The following functions are available for BunchKaufman objects: size, \, inv, issymmetric, ishermitian, getindex.

Examples

julia> A = [1 2; 2 3]
2×2 Array{Int64,2}:
1  2
2  3

julia> S = bunchkaufman(A) # A gets wrapped internally by Symmetric(A)
BunchKaufman{Float64,Array{Float64,2}}
D factor:
2×2 Tridiagonal{Float64,Array{Float64,1}}:
-0.333333  0.0
0.0       3.0
U factor:
2×2 UnitUpperTriangular{Float64,Array{Float64,2}}:
1.0  0.666667
⋅   1.0
permutation:
2-element Array{Int64,1}:
1
2

julia> d, u, p = S; # destructuring via iteration

julia> d == S.D && u == S.U && p == S.p
true

julia> S = bunchkaufman(Symmetric(A, :L))
BunchKaufman{Float64,Array{Float64,2}}
D factor:
2×2 Tridiagonal{Float64,Array{Float64,1}}:
3.0   0.0
0.0  -0.333333
L factor:
2×2 UnitLowerTriangular{Float64,Array{Float64,2}}:
1.0        ⋅
0.666667  1.0
permutation:
2-element Array{Int64,1}:
2
1
LinearAlgebra.eigvalsFunction
eigvals(A; permute::Bool=true, scale::Bool=true, sortby) -> values

Return the eigenvalues of A.

For general non-symmetric matrices it is possible to specify how the matrix is balanced before the eigenvalue calculation. The permute, scale, and sortby keywords are the same as for eigen!.

Examples

julia> diag_matrix = [1 0; 0 4]
2×2 Array{Int64,2}:
1  0
0  4

julia> eigvals(diag_matrix)
2-element Array{Float64,1}:
1.0
4.0

For a scalar input, eigvals will return a scalar.

Example

julia> eigvals(-2)
-2
eigvals(A, B) -> values

Computes the generalized eigenvalues of A and B.

Examples

julia> A = [1 0; 0 -1]
2×2 Array{Int64,2}:
1   0
0  -1

julia> B = [0 1; 1 0]
2×2 Array{Int64,2}:
0  1
1  0

julia> eigvals(A,B)
2-element Array{Complex{Float64},1}:
0.0 - 1.0im
0.0 + 1.0im
eigvals(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> values

Returns the eigenvalues of A. It is possible to calculate only a subset of the eigenvalues by specifying a UnitRange irange covering indices of the sorted eigenvalues, e.g. the 2nd to 8th eigenvalues.

julia> A = SymTridiagonal([1.; 2.; 1.], [2.; 3.])
3×3 SymTridiagonal{Float64,Array{Float64,1}}:
1.0  2.0   ⋅
2.0  2.0  3.0
⋅   3.0  1.0

julia> eigvals(A, 2:2)
1-element Array{Float64,1}:
0.9999999999999996

julia> eigvals(A)
3-element Array{Float64,1}:
-2.1400549446402604
1.0000000000000002
5.140054944640259
eigvals(A::Union{SymTridiagonal, Hermitian, Symmetric}, vl::Real, vu::Real) -> values

Returns the eigenvalues of A. It is possible to calculate only a subset of the eigenvalues by specifying a pair vl and vu for the lower and upper boundaries of the eigenvalues.

julia> A = SymTridiagonal([1.; 2.; 1.], [2.; 3.])
3×3 SymTridiagonal{Float64,Array{Float64,1}}:
1.0  2.0   ⋅
2.0  2.0  3.0
⋅   3.0  1.0

julia> eigvals(A, -1, 2)
1-element Array{Float64,1}:
1.0000000000000009

julia> eigvals(A)
3-element Array{Float64,1}:
-2.1400549446402604
1.0000000000000002
5.140054944640259
LinearAlgebra.eigvals!Function
eigvals!(A; permute::Bool=true, scale::Bool=true, sortby) -> values

Same as eigvals, but saves space by overwriting the input A, instead of creating a copy. The permute, scale, and sortby keywords are the same as for eigen.

Note

The input matrix A will not contain its eigenvalues after eigvals! is called on it - A is used as a workspace.

Examples

julia> A = [1. 2.; 3. 4.]
2×2 Array{Float64,2}:
1.0  2.0
3.0  4.0

julia> eigvals!(A)
2-element Array{Float64,1}:
-0.3722813232690143
5.372281323269014

julia> A
2×2 Array{Float64,2}:
-0.372281  -1.0
0.0        5.37228
eigvals!(A, B; sortby) -> values

Same as eigvals, but saves space by overwriting the input A (and B), instead of creating copies.

Note

The input matrices A and B will not contain their eigenvalues after eigvals! is called. They are used as workspaces.

Examples

julia> A = [1. 0.; 0. -1.]
2×2 Array{Float64,2}:
1.0   0.0
0.0  -1.0

julia> B = [0. 1.; 1. 0.]
2×2 Array{Float64,2}:
0.0  1.0
1.0  0.0

julia> eigvals!(A, B)
2-element Array{Complex{Float64},1}:
0.0 - 1.0im
0.0 + 1.0im

julia> A
2×2 Array{Float64,2}:
-0.0  -1.0
1.0  -0.0

julia> B
2×2 Array{Float64,2}:
1.0  0.0
0.0  1.0
eigvals!(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> values

Same as eigvals, but saves space by overwriting the input A, instead of creating a copy. irange is a range of eigenvalue indices to search for - for instance, the 2nd to 8th eigenvalues.

eigvals!(A::Union{SymTridiagonal, Hermitian, Symmetric}, vl::Real, vu::Real) -> values

Same as eigvals, but saves space by overwriting the input A, instead of creating a copy. vl is the lower bound of the interval to search for eigenvalues, and vu is the upper bound.

LinearAlgebra.eigmaxFunction
eigmax(A; permute::Bool=true, scale::Bool=true)

Return the largest eigenvalue of A. The option permute=true permutes the matrix to become closer to upper triangular, and scale=true scales the matrix by its diagonal elements to make rows and columns more equal in norm. Note that if the eigenvalues of A are complex, this method will fail, since complex numbers cannot be sorted.

Examples

julia> A = [0 im; -im 0]
2×2 Array{Complex{Int64},2}:
0+0im  0+1im
0-1im  0+0im

julia> eigmax(A)
1.0

julia> A = [0 im; -1 0]
2×2 Array{Complex{Int64},2}:
0+0im  0+1im
-1+0im  0+0im

julia> eigmax(A)
ERROR: DomainError with Complex{Int64}[0+0im 0+1im; -1+0im 0+0im]:
A cannot have complex eigenvalues.
Stacktrace:
[...]
LinearAlgebra.eigminFunction
eigmin(A; permute::Bool=true, scale::Bool=true)

Return the smallest eigenvalue of A. The option permute=true permutes the matrix to become closer to upper triangular, and scale=true scales the matrix by its diagonal elements to make rows and columns more equal in norm. Note that if the eigenvalues of A are complex, this method will fail, since complex numbers cannot be sorted.

Examples

julia> A = [0 im; -im 0]
2×2 Array{Complex{Int64},2}:
0+0im  0+1im
0-1im  0+0im

julia> eigmin(A)
-1.0

julia> A = [0 im; -1 0]
2×2 Array{Complex{Int64},2}:
0+0im  0+1im
-1+0im  0+0im

julia> eigmin(A)
ERROR: DomainError with Complex{Int64}[0+0im 0+1im; -1+0im 0+0im]:
A cannot have complex eigenvalues.
Stacktrace:
[...]
LinearAlgebra.eigvecsFunction
eigvecs(A::SymTridiagonal[, eigvals]) -> Matrix

Return a matrix M whose columns are the eigenvectors of A. (The kth eigenvector can be obtained from the slice M[:, k].)

If the optional vector of eigenvalues eigvals is specified, eigvecs returns the specific corresponding eigenvectors.

Examples

julia> A = SymTridiagonal([1.; 2.; 1.], [2.; 3.])
3×3 SymTridiagonal{Float64,Array{Float64,1}}:
1.0  2.0   ⋅
2.0  2.0  3.0
⋅   3.0  1.0

julia> eigvals(A)
3-element Array{Float64,1}:
-2.1400549446402604
1.0000000000000002
5.140054944640259

julia> eigvecs(A)
3×3 Array{Float64,2}:
0.418304  -0.83205      0.364299
-0.656749  -7.39009e-16  0.754109
0.627457   0.5547       0.546448

julia> eigvecs(A, [1.])
3×1 Array{Float64,2}:
0.8320502943378438
4.263514128092366e-17
-0.5547001962252291
eigvecs(A; permute::Bool=true, scale::Bool=true, sortby) -> Matrix

Return a matrix M whose columns are the eigenvectors of A. (The kth eigenvector can be obtained from the slice M[:, k].) The permute, scale, and sortby keywords are the same as for eigen.

Examples

julia> eigvecs([1.0 0.0 0.0; 0.0 3.0 0.0; 0.0 0.0 18.0])
3×3 Array{Float64,2}:
1.0  0.0  0.0
0.0  1.0  0.0
0.0  0.0  1.0
eigvecs(A, B) -> Matrix

Return a matrix M whose columns are the generalized eigenvectors of A and B. (The kth eigenvector can be obtained from the slice M[:, k].)

Examples

julia> A = [1 0; 0 -1]
2×2 Array{Int64,2}:
1   0
0  -1

julia> B = [0 1; 1 0]
2×2 Array{Int64,2}:
0  1
1  0

julia> eigvecs(A, B)
2×2 Array{Complex{Float64},2}:
0.0+1.0im   0.0-1.0im
-1.0+0.0im  -1.0-0.0im
LinearAlgebra.eigenFunction
eigen(A; permute::Bool=true, scale::Bool=true, sortby) -> Eigen

Computes the eigenvalue decomposition of A, returning an Eigen factorization object F which contains the eigenvalues in F.values and the eigenvectors in the columns of the matrix F.vectors. (The kth eigenvector can be obtained from the slice F.vectors[:, k].)

Iterating the decomposition produces the components F.values and F.vectors.

The following functions are available for Eigen objects: inv, det, and isposdef.

For general nonsymmetric matrices it is possible to specify how the matrix is balanced before the eigenvector calculation. The option permute=true permutes the matrix to become closer to upper triangular, and scale=true scales the matrix by its diagonal elements to make rows and columns more equal in norm. The default is true for both options.

By default, the eigenvalues and vectors are sorted lexicographically by (real(λ),imag(λ)). A different comparison function by(λ) can be passed to sortby, or you can pass sortby=nothing to leave the eigenvalues in an arbitrary order. Some special matrix types (e.g. Diagonal or SymTridiagonal) may implement their own sorting convention and not accept a sortby keyword.

Examples

julia> F = eigen([1.0 0.0 0.0; 0.0 3.0 0.0; 0.0 0.0 18.0])
Eigen{Float64,Float64,Array{Float64,2},Array{Float64,1}}
eigenvalues:
3-element Array{Float64,1}:
1.0
3.0
18.0
eigenvectors:
3×3 Array{Float64,2}:
1.0  0.0  0.0
0.0  1.0  0.0
0.0  0.0  1.0

julia> F.values
3-element Array{Float64,1}:
1.0
3.0
18.0

julia> F.vectors
3×3 Array{Float64,2}:
1.0  0.0  0.0
0.0  1.0  0.0
0.0  0.0  1.0

julia> vals, vecs = F; # destructuring via iteration

julia> vals == F.values && vecs == F.vectors
true
eigen(A, B) -> GeneralizedEigen

Computes the generalized eigenvalue decomposition of A and B, returning a GeneralizedEigen factorization object F which contains the generalized eigenvalues in F.values and the generalized eigenvectors in the columns of the matrix F.vectors. (The kth generalized eigenvector can be obtained from the slice F.vectors[:, k].)

Iterating the decomposition produces the components F.values and F.vectors.

Any keyword arguments passed to eigen are passed through to the lower-level eigen! function.

Examples

julia> A = [1 0; 0 -1]
2×2 Array{Int64,2}:
1   0
0  -1

julia> B = [0 1; 1 0]
2×2 Array{Int64,2}:
0  1
1  0

julia> F = eigen(A, B);

julia> F.values
2-element Array{Complex{Float64},1}:
0.0 - 1.0im
0.0 + 1.0im

julia> F.vectors
2×2 Array{Complex{Float64},2}:
0.0+1.0im   0.0-1.0im
-1.0+0.0im  -1.0-0.0im

julia> vals, vecs = F; # destructuring via iteration

julia> vals == F.values && vecs == F.vectors
true
eigen(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> Eigen

Computes the eigenvalue decomposition of A, returning an Eigen factorization object F which contains the eigenvalues in F.values and the eigenvectors in the columns of the matrix F.vectors. (The kth eigenvector can be obtained from the slice F.vectors[:, k].)

Iterating the decomposition produces the components F.values and F.vectors.

The following functions are available for Eigen objects: inv, det, and isposdef.

The UnitRange irange specifies indices of the sorted eigenvalues to search for.

Note

If irange is not 1:n, where n is the dimension of A, then the returned factorization will be a truncated factorization.

eigen(A::Union{SymTridiagonal, Hermitian, Symmetric}, vl::Real, vu::Real) -> Eigen

Computes the eigenvalue decomposition of A, returning an Eigen factorization object F which contains the eigenvalues in F.values and the eigenvectors in the columns of the matrix F.vectors. (The kth eigenvector can be obtained from the slice F.vectors[:, k].)

Iterating the decomposition produces the components F.values and F.vectors.

The following functions are available for Eigen objects: inv, det, and isposdef.

vl is the lower bound of the window of eigenvalues to search for, and vu is the upper bound.

Note

If [vl, vu] does not contain all eigenvalues of A, then the returned factorization will be a truncated factorization.

LinearAlgebra.hessenbergFunction
hessenberg(A) -> Hessenberg

Compute the Hessenberg decomposition of A and return a Hessenberg object. If F is the factorization object, the unitary matrix can be accessed with F.Q (of type LinearAlgebra.HessenbergQ) and the Hessenberg matrix with F.H (of type UpperHessenberg), either of which may be converted to a regular matrix with Matrix(F.H) or Matrix(F.Q).

If A is Hermitian or real-Symmetric, then the Hessenberg decomposition produces a real-symmetric tridiagonal matrix and F.H is of type SymTridiagonal.

Note that the shifted factorization A+μI = Q (H+μI) Q' can be constructed efficiently by F + μ*I using the UniformScaling object I, which creates a new Hessenberg object with shared storage and a modified shift. The shift of a given F is obtained by F.μ. This is useful because multiple shifted solves (F + μ*I) \ b (for different μ and/or b) can be performed efficiently once F is created.

Iterating the decomposition produces the factors F.Q, F.H, F.μ.

Examples

julia> A = [4. 9. 7.; 4. 4. 1.; 4. 3. 2.]
3×3 Array{Float64,2}:
4.0  9.0  7.0
4.0  4.0  1.0
4.0  3.0  2.0

julia> F = hessenberg(A);

julia> F.Q * F.H * F.Q'
3×3 Array{Float64,2}:
4.0  9.0  7.0
4.0  4.0  1.0
4.0  3.0  2.0

julia> q, h = F; # destructuring via iteration

julia> q == F.Q && h == F.H
true
LinearAlgebra.schur!Function
schur!(A::StridedMatrix) -> F::Schur

Same as schur but uses the input argument A as workspace.

Examples

julia> A = [5. 7.; -2. -4.]
2×2 Array{Float64,2}:
5.0   7.0
-2.0  -4.0

julia> F = schur!(A)
Schur{Float64,Array{Float64,2}}
T factor:
2×2 Array{Float64,2}:
3.0   9.0
0.0  -2.0
Z factor:
2×2 Array{Float64,2}:
0.961524  0.274721
-0.274721  0.961524
eigenvalues:
2-element Array{Float64,1}:
3.0
-2.0

julia> A
2×2 Array{Float64,2}:
3.0   9.0
0.0  -2.0
schur!(A::StridedMatrix, B::StridedMatrix) -> F::GeneralizedSchur

Same as schur but uses the input matrices A and B as workspace.

LinearAlgebra.schurFunction
schur(A::StridedMatrix) -> F::Schur

Computes the Schur factorization of the matrix A. The (quasi) triangular Schur factor can be obtained from the Schur object F with either F.Schur or F.T and the orthogonal/unitary Schur vectors can be obtained with F.vectors or F.Z such that A = F.vectors * F.Schur * F.vectors'. The eigenvalues of A can be obtained with F.values.

Iterating the decomposition produces the components F.T, F.Z, and F.values.

Examples

julia> A = [5. 7.; -2. -4.]
2×2 Array{Float64,2}:
5.0   7.0
-2.0  -4.0

julia> F = schur(A)
Schur{Float64,Array{Float64,2}}
T factor:
2×2 Array{Float64,2}:
3.0   9.0
0.0  -2.0
Z factor:
2×2 Array{Float64,2}:
0.961524  0.274721
-0.274721  0.961524
eigenvalues:
2-element Array{Float64,1}:
3.0
-2.0

julia> F.vectors * F.Schur * F.vectors'
2×2 Array{Float64,2}:
5.0   7.0
-2.0  -4.0

julia> t, z, vals = F; # destructuring via iteration

julia> t == F.T && z == F.Z && vals == F.values
true
schur(A::StridedMatrix, B::StridedMatrix) -> F::GeneralizedSchur

Computes the Generalized Schur (or QZ) factorization of the matrices A and B. The (quasi) triangular Schur factors can be obtained from the Schur object F with F.S and F.T, the left unitary/orthogonal Schur vectors can be obtained with F.left or F.Q and the right unitary/orthogonal Schur vectors can be obtained with F.right or F.Z such that A=F.left*F.S*F.right' and B=F.left*F.T*F.right'. The generalized eigenvalues of A and B can be obtained with F.α./F.β.

Iterating the decomposition produces the components F.S, F.T, F.Q, F.Z, F.α, and F.β.

LinearAlgebra.ordschurFunction
ordschur(F::Schur, select::Union{Vector{Bool},BitVector}) -> F::Schur

Reorders the Schur factorization F of a matrix A = Z*T*Z' according to the logical array select returning the reordered factorization F object. The selected eigenvalues appear in the leading diagonal of F.Schur and the corresponding leading columns of F.vectors form an orthogonal/unitary basis of the corresponding right invariant subspace. In the real case, a complex conjugate pair of eigenvalues must be either both included or both excluded via select.

ordschur(F::GeneralizedSchur, select::Union{Vector{Bool},BitVector}) -> F::GeneralizedSchur

Reorders the Generalized Schur factorization F of a matrix pair (A, B) = (Q*S*Z', Q*T*Z') according to the logical array select and returns a GeneralizedSchur object F. The selected eigenvalues appear in the leading diagonal of both F.S and F.T, and the left and right orthogonal/unitary Schur vectors are also reordered such that (A, B) = F.Q*(F.S, F.T)*F.Z' still holds and the generalized eigenvalues of A and B can still be obtained with F.α./F.β.

LinearAlgebra.ordschur!Function
ordschur!(F::Schur, select::Union{Vector{Bool},BitVector}) -> F::Schur

Same as ordschur but overwrites the factorization F.

ordschur!(F::GeneralizedSchur, select::Union{Vector{Bool},BitVector}) -> F::GeneralizedSchur

Same as ordschur but overwrites the factorization F.

LinearAlgebra.svdFunction
svd(A; full::Bool = false, alg::Algorithm = default_svd_alg(A)) -> SVD

Compute the singular value decomposition (SVD) of A and return an SVD object.

U, S, V and Vt can be obtained from the factorization F with F.U, F.S, F.V and F.Vt, such that A = U * Diagonal(S) * Vt. The algorithm produces Vt and hence Vt is more efficient to extract than V. The singular values in S are sorted in descending order.

Iterating the decomposition produces the components U, S, and V.

If full = false (default), a "thin" SVD is returned. For a $M \times N$ matrix A, in the full factorization U is M \times M and V is N \times N, while in the thin factorization U is M \times K and V is N \times K, where K = \min(M,N) is the number of singular values.

If alg = DivideAndConquer() a divide-and-conquer algorithm is used to calculate the SVD. Another (typically slower but more accurate) option is alg = QRIteration().

Julia 1.3

The alg keyword argument requires Julia 1.3 or later.

Examples

julia> A = [1. 0. 0. 0. 2.; 0. 0. 3. 0. 0.; 0. 0. 0. 0. 0.; 0. 2. 0. 0. 0.]
4×5 Array{Float64,2}:
1.0  0.0  0.0  0.0  2.0
0.0  0.0  3.0  0.0  0.0
0.0  0.0  0.0  0.0  0.0
0.0  2.0  0.0  0.0  0.0

julia> F = svd(A);

julia> F.U * Diagonal(F.S) * F.Vt
4×5 Array{Float64,2}:
1.0  0.0  0.0  0.0  2.0
0.0  0.0  3.0  0.0  0.0
0.0  0.0  0.0  0.0  0.0
0.0  2.0  0.0  0.0  0.0

julia> u, s, v = F; # destructuring via iteration

julia> u == F.U && s == F.S && v == F.V
true
svd(A, B) -> GeneralizedSVD

Compute the generalized SVD of A and B, returning a GeneralizedSVD factorization object F, such that A = F.U*F.D1*F.R0*F.Q' and B = F.V*F.D2*F.R0*F.Q'.

For an M-by-N matrix A and P-by-N matrix B,

• U is a M-by-M orthogonal matrix,
• V is a P-by-P orthogonal matrix,
• Q is a N-by-N orthogonal matrix,
• D1 is a M-by-(K+L) diagonal matrix with 1s in the first K entries,
• D2 is a P-by-(K+L) matrix whose top right L-by-L block is diagonal,
• R0 is a (K+L)-by-N matrix whose rightmost (K+L)-by-(K+L) block is nonsingular upper block triangular,

K+L is the effective numerical rank of the matrix [A; B].

Iterating the decomposition produces the components U, V, Q, D1, D2, and R0.

The entries of F.D1 and F.D2 are related, as explained in the LAPACK documentation for the generalized SVD and the xGGSVD3 routine which is called underneath (in LAPACK 3.6.0 and newer).

Examples

julia> A = [1. 0.; 0. -1.]
2×2 Array{Float64,2}:
1.0   0.0
0.0  -1.0

julia> B = [0. 1.; 1. 0.]
2×2 Array{Float64,2}:
0.0  1.0
1.0  0.0

julia> F = svd(A, B);

julia> F.U*F.D1*F.R0*F.Q'
2×2 Array{Float64,2}:
1.0   0.0
0.0  -1.0

julia> F.V*F.D2*F.R0*F.Q'
2×2 Array{Float64,2}:
0.0  1.0
1.0  0.0
LinearAlgebra.svd!Function
svd!(A; full::Bool = false, alg::Algorithm = default_svd_alg(A)) -> SVD

svd! is the same as svd, but saves space by overwriting the input A, instead of creating a copy.

Examples

julia> A = [1. 0. 0. 0. 2.; 0. 0. 3. 0. 0.; 0. 0. 0. 0. 0.; 0. 2. 0. 0. 0.]
4×5 Array{Float64,2}:
1.0  0.0  0.0  0.0  2.0
0.0  0.0  3.0  0.0  0.0
0.0  0.0  0.0  0.0  0.0
0.0  2.0  0.0  0.0  0.0

julia> F = svd!(A);

julia> F.U * Diagonal(F.S) * F.Vt
4×5 Array{Float64,2}:
1.0  0.0  0.0  0.0  2.0
0.0  0.0  3.0  0.0  0.0
0.0  0.0  0.0  0.0  0.0
0.0  2.0  0.0  0.0  0.0

julia> A
4×5 Array{Float64,2}:
-2.23607   0.0   0.0  0.0  0.618034
0.0      -3.0   1.0  0.0  0.0
0.0       0.0   0.0  0.0  0.0
0.0       0.0  -2.0  0.0  0.0
svd!(A, B) -> GeneralizedSVD

svd! is the same as svd, but modifies the arguments A and B in-place, instead of making copies.

Examples

julia> A = [1. 0.; 0. -1.]
2×2 Array{Float64,2}:
1.0   0.0
0.0  -1.0

julia> B = [0. 1.; 1. 0.]
2×2 Array{Float64,2}:
0.0  1.0
1.0  0.0

julia> F = svd!(A, B);

julia> F.U*F.D1*F.R0*F.Q'
2×2 Array{Float64,2}:
1.0   0.0
0.0  -1.0

julia> F.V*F.D2*F.R0*F.Q'
2×2 Array{Float64,2}:
0.0  1.0
1.0  0.0

julia> A
2×2 Array{Float64,2}:
1.41421   0.0
0.0      -1.41421

julia> B
2×2 Array{Float64,2}:
1.0  -0.0
0.0  -1.0
LinearAlgebra.svdvalsFunction
svdvals(A)

Return the singular values of A in descending order.

Examples

julia> A = [1. 0. 0. 0. 2.; 0. 0. 3. 0. 0.; 0. 0. 0. 0. 0.; 0. 2. 0. 0. 0.]
4×5 Array{Float64,2}:
1.0  0.0  0.0  0.0  2.0
0.0  0.0  3.0  0.0  0.0
0.0  0.0  0.0  0.0  0.0
0.0  2.0  0.0  0.0  0.0

julia> svdvals(A)
4-element Array{Float64,1}:
3.0
2.23606797749979
2.0
0.0
svdvals(A, B)

Return the generalized singular values from the generalized singular value decomposition of A and B. See also svd.

Examples

julia> A = [1. 0.; 0. -1.]
2×2 Array{Float64,2}:
1.0   0.0
0.0  -1.0

julia> B = [0. 1.; 1. 0.]
2×2 Array{Float64,2}:
0.0  1.0
1.0  0.0

julia> svdvals(A, B)
2-element Array{Float64,1}:
1.0
1.0
LinearAlgebra.svdvals!Function
svdvals!(A)

Return the singular values of A, saving space by overwriting the input. See also svdvals and svd.

Examples

julia> A = [1. 0. 0. 0. 2.; 0. 0. 3. 0. 0.; 0. 0. 0. 0. 0.; 0. 2. 0. 0. 0.]
4×5 Array{Float64,2}:
1.0  0.0  0.0  0.0  2.0
0.0  0.0  3.0  0.0  0.0
0.0  0.0  0.0  0.0  0.0
0.0  2.0  0.0  0.0  0.0

julia> svdvals!(A)
4-element Array{Float64,1}:
3.0
2.23606797749979
2.0
0.0

julia> A
4×5 Array{Float64,2}:
-2.23607   0.0   0.0  0.0  0.618034
0.0      -3.0   1.0  0.0  0.0
0.0       0.0   0.0  0.0  0.0
0.0       0.0  -2.0  0.0  0.0
svdvals!(A, B)

Return the generalized singular values from the generalized singular value decomposition of A and B, saving space by overwriting A and B. See also svd and svdvals.

Examples

julia> A = [1. 0.; 0. -1.]
2×2 Array{Float64,2}:
1.0   0.0
0.0  -1.0

julia> B = [0. 1.; 1. 0.]
2×2 Array{Float64,2}:
0.0  1.0
1.0  0.0

julia> svdvals!(A, B)
2-element Array{Float64,1}:
1.0
1.0

julia> A
2×2 Array{Float64,2}:
1.41421   0.0
0.0      -1.41421

julia> B
2×2 Array{Float64,2}:
1.0  -0.0
0.0  -1.0
LinearAlgebra.GivensType
LinearAlgebra.Givens(i1,i2,c,s) -> G

A Givens rotation linear operator. The fields c and s represent the cosine and sine of the rotation angle, respectively. The Givens type supports left multiplication G*A and conjugated transpose right multiplication A*G'. The type doesn't have a size and can therefore be multiplied with matrices of arbitrary size as long as i2<=size(A,2) for G*A or i2<=size(A,1) for A*G'.

See also: givens

LinearAlgebra.givensFunction
givens(f::T, g::T, i1::Integer, i2::Integer) where {T} -> (G::Givens, r::T)

Computes the Givens rotation G and scalar r such that for any vector x where

x[i1] = f
x[i2] = g

the result of the multiplication

y = G*x

has the property that

y[i1] = r
y[i2] = 0

See also: LinearAlgebra.Givens

givens(A::AbstractArray, i1::Integer, i2::Integer, j::Integer) -> (G::Givens, r)

Computes the Givens rotation G and scalar r such that the result of the multiplication

B = G*A

has the property that

B[i1,j] = r
B[i2,j] = 0

See also: LinearAlgebra.Givens

givens(x::AbstractVector, i1::Integer, i2::Integer) -> (G::Givens, r)

Computes the Givens rotation G and scalar r such that the result of the multiplication

B = G*x

has the property that

B[i1] = r
B[i2] = 0

See also: LinearAlgebra.Givens

LinearAlgebra.triuFunction
triu(M)

Upper triangle of a matrix.

Examples

julia> a = fill(1.0, (4,4))
4×4 Array{Float64,2}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0

julia> triu(a)
4×4 Array{Float64,2}:
1.0  1.0  1.0  1.0
0.0  1.0  1.0  1.0
0.0  0.0  1.0  1.0
0.0  0.0  0.0  1.0
triu(M, k::Integer)

Returns the upper triangle of M starting from the kth superdiagonal.

Examples

julia> a = fill(1.0, (4,4))
4×4 Array{Float64,2}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0

julia> triu(a,3)
4×4 Array{Float64,2}:
0.0  0.0  0.0  1.0
0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0

julia> triu(a,-3)
4×4 Array{Float64,2}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
LinearAlgebra.triu!Function
triu!(M)

Upper triangle of a matrix, overwriting M in the process. See also triu.

triu!(M, k::Integer)

Return the upper triangle of M starting from the kth superdiagonal, overwriting M in the process.

Examples

julia> M = [1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5]
5×5 Array{Int64,2}:
1  2  3  4  5
1  2  3  4  5
1  2  3  4  5
1  2  3  4  5
1  2  3  4  5

julia> triu!(M, 1)
5×5 Array{Int64,2}:
0  2  3  4  5
0  0  3  4  5
0  0  0  4  5
0  0  0  0  5
0  0  0  0  0
LinearAlgebra.trilFunction
tril(M)

Lower triangle of a matrix.

Examples

julia> a = fill(1.0, (4,4))
4×4 Array{Float64,2}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0

julia> tril(a)
4×4 Array{Float64,2}:
1.0  0.0  0.0  0.0
1.0  1.0  0.0  0.0
1.0  1.0  1.0  0.0
1.0  1.0  1.0  1.0
tril(M, k::Integer)

Returns the lower triangle of M starting from the kth superdiagonal.

Examples

julia> a = fill(1.0, (4,4))
4×4 Array{Float64,2}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0

julia> tril(a,3)
4×4 Array{Float64,2}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0

julia> tril(a,-3)
4×4 Array{Float64,2}:
0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0
1.0  0.0  0.0  0.0
LinearAlgebra.tril!Function
tril!(M)

Lower triangle of a matrix, overwriting M in the process. See also tril.

tril!(M, k::Integer)

Return the lower triangle of M starting from the kth superdiagonal, overwriting M in the process.

Examples

julia> M = [1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5]
5×5 Array{Int64,2}:
1  2  3  4  5
1  2  3  4  5
1  2  3  4  5
1  2  3  4  5
1  2  3  4  5

julia> tril!(M, 2)
5×5 Array{Int64,2}:
1  2  3  0  0
1  2  3  4  0
1  2  3  4  5
1  2  3  4  5
1  2  3  4  5
LinearAlgebra.diagindFunction
diagind(M, k::Integer=0)

An AbstractRange giving the indices of the kth diagonal of the matrix M.

Examples

julia> A = [1 2 3; 4 5 6; 7 8 9]
3×3 Array{Int64,2}:
1  2  3
4  5  6
7  8  9

julia> diagind(A,-1)
2:4:6
LinearAlgebra.diagFunction
diag(M, k::Integer=0)

The kth diagonal of a matrix, as a vector.

See also: diagm

Examples

julia> A = [1 2 3; 4 5 6; 7 8 9]
3×3 Array{Int64,2}:
1  2  3
4  5  6
7  8  9

julia> diag(A,1)
2-element Array{Int64,1}:
2
6
LinearAlgebra.diagmFunction
diagm(kv::Pair{<:Integer,<:AbstractVector}...)
diagm(m::Integer, n::Integer, kv::Pair{<:Integer,<:AbstractVector}...)

Construct a matrix from Pairs of diagonals and vectors. Vector kv.second will be placed on the kv.first diagonal. By default the matrix is square and its size is inferred from kv, but a non-square size m×n (padded with zeros as needed) can be specified by passing m,n as the first arguments.

diagm constructs a full matrix; if you want storage-efficient versions with fast arithmetic, see Diagonal, Bidiagonal Tridiagonal and SymTridiagonal.

Examples

julia> diagm(1 => [1,2,3])
4×4 Array{Int64,2}:
0  1  0  0
0  0  2  0
0  0  0  3
0  0  0  0

julia> diagm(1 => [1,2,3], -1 => [4,5])
4×4 Array{Int64,2}:
0  1  0  0
4  0  2  0
0  5  0  3
0  0  0  0
diagm(v::AbstractVector)
diagm(m::Integer, n::Integer, v::AbstractVector)

Construct a matrix with elements of the vector as diagonal elements. By default (if size=nothing), the matrix is square and its size is given by length(v), but a non-square size m×n can be specified by passing m,n as the first arguments.

Examples

julia> diagm([1,2,3])
3×3 Array{Int64,2}:
1  0  0
0  2  0
0  0  3
LinearAlgebra.rankFunction
rank(A::AbstractMatrix; atol::Real=0, rtol::Real=atol>0 ? 0 : n*ϵ)
rank(A::AbstractMatrix, rtol::Real)

Compute the rank of a matrix by counting how many singular values of A have magnitude greater than max(atol, rtol*σ₁) where σ₁ is A's largest singular value. atol and rtol are the absolute and relative tolerances, respectively. The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of A, and ϵ is the eps of the element type of A.

Julia 1.1

The atol and rtol keyword arguments requires at least Julia 1.1. In Julia 1.0 rtol is available as a positional argument, but this will be deprecated in Julia 2.0.

Examples

julia> rank(Matrix(I, 3, 3))
3

julia> rank(diagm(0 => [1, 0, 2]))
2

julia> rank(diagm(0 => [1, 0.001, 2]), rtol=0.1)
2

julia> rank(diagm(0 => [1, 0.001, 2]), rtol=0.00001)
3

julia> rank(diagm(0 => [1, 0.001, 2]), atol=1.5)
1
LinearAlgebra.normFunction
norm(A, p::Real=2)

For any iterable container A (including arrays of any dimension) of numbers (or any element type for which norm is defined), compute the p-norm (defaulting to p=2) as if A were a vector of the corresponding length.

The p-norm is defined as

$\|A\|_p = \left( \sum_{i=1}^n | a_i | ^p \right)^{1/p}$

with $a_i$ the entries of $A$, $| a_i |$ the norm of $a_i$, and $n$ the length of $A$. Since the p-norm is computed using the norms of the entries of A, the p-norm of a vector of vectors is not compatible with the interpretation of it as a block vector in general if p != 2.

p can assume any numeric value (even though not all values produce a mathematically valid vector norm). In particular, norm(A, Inf) returns the largest value in abs.(A), whereas norm(A, -Inf) returns the smallest. If A is a matrix and p=2, then this is equivalent to the Frobenius norm.

The second argument p is not necessarily a part of the interface for norm, i.e. a custom type may only implement norm(A) without second argument.

Use opnorm to compute the operator norm of a matrix.

Examples

julia> v = [3, -2, 6]
3-element Array{Int64,1}:
3
-2
6

julia> norm(v)
7.0

julia> norm(v, 1)
11.0

julia> norm(v, Inf)
6.0

julia> norm([1 2 3; 4 5 6; 7 8 9])
16.881943016134134

julia> norm([1 2 3 4 5 6 7 8 9])
16.881943016134134

julia> norm(1:9)
16.881943016134134

julia> norm(hcat(v,v), 1) == norm(vcat(v,v), 1) != norm([v,v], 1)
true

julia> norm(hcat(v,v), 2) == norm(vcat(v,v), 2) == norm([v,v], 2)
true

julia> norm(hcat(v,v), Inf) == norm(vcat(v,v), Inf) != norm([v,v], Inf)
true
norm(x::Number, p::Real=2)

For numbers, return $\left( |x|^p \right)^{1/p}$.

Examples

julia> norm(2, 1)
2.0

julia> norm(-2, 1)
2.0

julia> norm(2, 2)
2.0

julia> norm(-2, 2)
2.0

julia> norm(2, Inf)
2.0

julia> norm(-2, Inf)
2.0
LinearAlgebra.opnormFunction
opnorm(A::AbstractMatrix, p::Real=2)

Compute the operator norm (or matrix norm) induced by the vector p-norm, where valid values of p are 1, 2, or Inf. (Note that for sparse matrices, p=2 is currently not implemented.) Use norm to compute the Frobenius norm.

When p=1, the operator norm is the maximum absolute column sum of A:

$\|A\|_1 = \max_{1 ≤ j ≤ n} \sum_{i=1}^m | a_{ij} |$

with $a_{ij}$ the entries of $A$, and $m$ and $n$ its dimensions.

When p=2, the operator norm is the spectral norm, equal to the largest singular value of A.

When p=Inf, the operator norm is the maximum absolute row sum of A:

$\|A\|_\infty = \max_{1 ≤ i ≤ m} \sum _{j=1}^n | a_{ij} |$

Examples

julia> A = [1 -2 -3; 2 3 -1]
2×3 Array{Int64,2}:
1  -2  -3
2   3  -1

julia> opnorm(A, Inf)
6.0

julia> opnorm(A, 1)
5.0
opnorm(x::Number, p::Real=2)

For numbers, return $\left( |x|^p \right)^{1/p}$. This is equivalent to norm.

opnorm(A::Adjoint{<:Any,<:AbstracVector}, q::Real=2)
opnorm(A::Transpose{<:Any,<:AbstracVector}, q::Real=2)

For Adjoint/Transpose-wrapped vectors, return the operator $q$-norm of A, which is equivalent to the p-norm with value p = q/(q-1). They coincide at p = q = 2. Use norm to compute the p norm of A as a vector.

The difference in norm between a vector space and its dual arises to preserve the relationship between duality and the dot product, and the result is consistent with the operator p-norm of a 1 × n matrix.

Examples

julia> v = [1; im];

julia> vc = v';

julia> opnorm(vc, 1)
1.0

julia> norm(vc, 1)
2.0

julia> norm(v, 1)
2.0

julia> opnorm(vc, 2)
1.4142135623730951

julia> norm(vc, 2)
1.4142135623730951

julia> norm(v, 2)
1.4142135623730951

julia> opnorm(vc, Inf)
2.0

julia> norm(vc, Inf)
1.0

julia> norm(v, Inf)
1.0
LinearAlgebra.normalizeFunction
normalize(v::AbstractVector, p::Real=2)

Normalize the vector v so that its p-norm equals unity, i.e. norm(v, p) == 1. See also normalize! and norm.

Examples

julia> a = [1,2,4];

julia> b = normalize(a)
3-element Array{Float64,1}:
0.2182178902359924
0.4364357804719848
0.8728715609439696

julia> norm(b)
1.0

julia> c = normalize(a, 1)
3-element Array{Float64,1}:
0.14285714285714285
0.2857142857142857
0.5714285714285714

julia> norm(c, 1)
1.0
LinearAlgebra.condFunction
cond(M, p::Real=2)

Condition number of the matrix M, computed using the operator p-norm. Valid values for p are 1, 2 (default), or Inf.

LinearAlgebra.condskeelFunction
condskeel(M, [x, p::Real=Inf])
$\kappa_S(M, p) = \left\Vert \left\vert M \right\vert \left\vert M^{-1} \right\vert \right\Vert_p \\ \kappa_S(M, x, p) = \left\Vert \left\vert M \right\vert \left\vert M^{-1} \right\vert \left\vert x \right\vert \right\Vert_p$

Skeel condition number $\kappa_S$ of the matrix M, optionally with respect to the vector x, as computed using the operator p-norm. $\left\vert M \right\vert$ denotes the matrix of (entry wise) absolute values of $M$; $\left\vert M \right\vert_{ij} = \left\vert M_{ij} \right\vert$. Valid values for p are 1, 2 and Inf (default).

This quantity is also known in the literature as the Bauer condition number, relative condition number, or componentwise relative condition number.

LinearAlgebra.trFunction
tr(M)

Matrix trace. Sums the diagonal elements of M.

Examples

julia> A = [1 2; 3 4]
2×2 Array{Int64,2}:
1  2
3  4

julia> tr(A)
5
LinearAlgebra.detFunction
det(M)

Matrix determinant.

Examples

julia> M = [1 0; 2 2]
2×2 Array{Int64,2}:
1  0
2  2

julia> det(M)
2.0
LinearAlgebra.logdetFunction
logdet(M)

Log of matrix determinant. Equivalent to log(det(M)), but may provide increased accuracy and/or speed.

Examples

julia> M = [1 0; 2 2]
2×2 Array{Int64,2}:
1  0
2  2

julia> logdet(M)
0.6931471805599453

julia> logdet(Matrix(I, 3, 3))
0.0
LinearAlgebra.logabsdetFunction
logabsdet(M)

Log of absolute value of matrix determinant. Equivalent to (log(abs(det(M))), sign(det(M))), but may provide increased accuracy and/or speed.

Examples

julia> A = [-1. 0.; 0. 1.]
2×2 Array{Float64,2}:
-1.0  0.0
0.0  1.0

julia> det(A)
-1.0

julia> logabsdet(A)
(0.0, -1.0)

julia> B = [2. 0.; 0. 1.]
2×2 Array{Float64,2}:
2.0  0.0
0.0  1.0

julia> det(B)
2.0

julia> logabsdet(B)
(0.6931471805599453, 1.0)
Base.invMethod
inv(M)

Matrix inverse. Computes matrix N such that M * N = I, where I is the identity matrix. Computed by solving the left-division N = M \ I.

Examples

julia> M = [2 5; 1 3]
2×2 Array{Int64,2}:
2  5
1  3

julia> N = inv(M)
2×2 Array{Float64,2}:
3.0  -5.0
-1.0   2.0

julia> M*N == N*M == Matrix(I, 2, 2)
true
LinearAlgebra.pinvFunction
pinv(M; atol::Real=0, rtol::Real=atol>0 ? 0 : n*ϵ)
pinv(M, rtol::Real) = pinv(M; rtol=rtol) # to be deprecated in Julia 2.0

Computes the Moore-Penrose pseudoinverse.

For matrices M with floating point elements, it is convenient to compute the pseudoinverse by inverting only singular values greater than max(atol, rtol*σ₁) where σ₁ is the largest singular value of M.

The optimal choice of absolute (atol) and relative tolerance (rtol) varies both with the value of M and the intended application of the pseudoinverse. The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of M, and ϵ is the eps of the element type of M.

For inverting dense ill-conditioned matrices in a least-squares sense, rtol = sqrt(eps(real(float(one(eltype(M)))))) is recommended.

For more information, see [issue8859], [B96], [S84], [KY88].

Examples

julia> M = [1.5 1.3; 1.2 1.9]
2×2 Array{Float64,2}:
1.5  1.3
1.2  1.9

julia> N = pinv(M)
2×2 Array{Float64,2}:
1.47287   -1.00775
-0.930233   1.16279

julia> M * N
2×2 Array{Float64,2}:
1.0          -2.22045e-16
4.44089e-16   1.0
LinearAlgebra.nullspaceFunction
nullspace(M; atol::Real=0, rtol::Rea=atol>0 ? 0 : n*ϵ)
nullspace(M, rtol::Real) = nullspace(M; rtol=rtol) # to be deprecated in Julia 2.0

Computes a basis for the nullspace of M by including the singular vectors of A whose singular have magnitude are greater than max(atol, rtol*σ₁), where σ₁ is M's largest singularvalue.

By default, the relative tolerance rtol is n*ϵ, where n is the size of the smallest dimension of M, and ϵ is the eps of the element type of M.

Examples

julia> M = [1 0 0; 0 1 0; 0 0 0]
3×3 Array{Int64,2}:
1  0  0
0  1  0
0  0  0

julia> nullspace(M)
3×1 Array{Float64,2}:
0.0
0.0
1.0

julia> nullspace(M, rtol=3)
3×3 Array{Float64,2}:
0.0  1.0  0.0
1.0  0.0  0.0
0.0  0.0  1.0

julia> nullspace(M, atol=0.95)
3×1 Array{Float64,2}:
0.0
0.0
1.0
Base.kronFunction
kron(A, B)

Kronecker tensor product of two vectors or two matrices.

For vectors v and w, the Kronecker product is related to the outer product by kron(v,w) == vec(w*transpose(v)) or w*transpose(v) == reshape(kron(v,w), (length(w), length(v))). Note how the ordering of v and w differs on the left and right of these expressions (due to column-major storage).

Examples

julia> A = [1 2; 3 4]
2×2 Array{Int64,2}:
1  2
3  4

julia> B = [im 1; 1 -im]
2×2 Array{Complex{Int64},2}:
0+1im  1+0im
1+0im  0-1im

julia> kron(A, B)
4×4 Array{Complex{Int64},2}:
0+1im  1+0im  0+2im  2+0im
1+0im  0-1im  2+0im  0-2im
0+3im  3+0im  0+4im  4+0im
3+0im  0-3im  4+0im  0-4im

julia> v = [1, 2]; w = [3, 4, 5];

julia> w*transpose(v)
3×2 Array{Int64,2}:
3   6
4   8
5  10

julia> reshape(kron(v,w), (length(w), length(v)))
3×2 Array{Int64,2}:
3   6
4   8
5  10
Base.expMethod
exp(A::AbstractMatrix)

Compute the matrix exponential of A, defined by

$e^A = \sum_{n=0}^{\infty} \frac{A^n}{n!}.$

For symmetric or Hermitian A, an eigendecomposition (eigen) is used, otherwise the scaling and squaring algorithm (see [H05]) is chosen.

Examples

julia> A = Matrix(1.0I, 2, 2)
2×2 Array{Float64,2}:
1.0  0.0
0.0  1.0

julia> exp(A)
2×2 Array{Float64,2}:
2.71828  0.0
0.0      2.71828
Base.logMethod
log(A{T}::StridedMatrix{T})

If A has no negative real eigenvalue, compute the principal matrix logarithm of A, i.e. the unique matrix $X$ such that $e^X = A$ and $-\pi < Im(\lambda) < \pi$ for all the eigenvalues $\lambda$ of $X$. If A has nonpositive eigenvalues, a nonprincipal matrix function is returned whenever possible.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used, if A is triangular an improved version of the inverse scaling and squaring method is employed (see [AH12] and [AHR13]). For general matrices, the complex Schur form (schur) is computed and the triangular algorithm is used on the triangular factor.

Examples

julia> A = Matrix(2.7182818*I, 2, 2)
2×2 Array{Float64,2}:
2.71828  0.0
0.0      2.71828

julia> log(A)
2×2 Array{Float64,2}:
1.0  0.0
0.0  1.0
Base.sqrtMethod
sqrt(A::AbstractMatrix)

If A has no negative real eigenvalues, compute the principal matrix square root of A, that is the unique matrix $X$ with eigenvalues having positive real part such that $X^2 = A$. Otherwise, a nonprincipal square root is returned.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the square root. Otherwise, the square root is determined by means of the Björck-Hammarling method [BH83], which computes the complex Schur form (schur) and then the complex square root of the triangular factor.

Examples

julia> A = [4 0; 0 4]
2×2 Array{Int64,2}:
4  0
0  4

julia> sqrt(A)
2×2 Array{Float64,2}:
2.0  0.0
0.0  2.0
Base.cosMethod
cos(A::AbstractMatrix)

Compute the matrix cosine of a square matrix A.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the cosine. Otherwise, the cosine is determined by calling exp.

Examples

julia> cos(fill(1.0, (2,2)))
2×2 Array{Float64,2}:
0.291927  -0.708073
-0.708073   0.291927
Base.sinMethod
sin(A::AbstractMatrix)

Compute the matrix sine of a square matrix A.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the sine. Otherwise, the sine is determined by calling exp.

Examples

julia> sin(fill(1.0, (2,2)))
2×2 Array{Float64,2}:
0.454649  0.454649
0.454649  0.454649
Base.Math.sincosMethod
sincos(A::AbstractMatrix)

Compute the matrix sine and cosine of a square matrix A.

Examples

julia> S, C = sincos(fill(1.0, (2,2)));

julia> S
2×2 Array{Float64,2}:
0.454649  0.454649
0.454649  0.454649

julia> C
2×2 Array{Float64,2}:
0.291927  -0.708073
-0.708073   0.291927
Base.tanMethod
tan(A::AbstractMatrix)

Compute the matrix tangent of a square matrix A.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the tangent. Otherwise, the tangent is determined by calling exp.

Examples

julia> tan(fill(1.0, (2,2)))
2×2 Array{Float64,2}:
-1.09252  -1.09252
-1.09252  -1.09252
Base.Math.secMethod
sec(A::AbstractMatrix)

Compute the matrix secant of a square matrix A.

Base.Math.cscMethod
csc(A::AbstractMatrix)

Compute the matrix cosecant of a square matrix A.

Base.Math.cotMethod
cot(A::AbstractMatrix)

Compute the matrix cotangent of a square matrix A.

Base.coshMethod
cosh(A::AbstractMatrix)

Compute the matrix hyperbolic cosine of a square matrix A.

Base.sinhMethod
sinh(A::AbstractMatrix)

Compute the matrix hyperbolic sine of a square matrix A.

Base.tanhMethod
tanh(A::AbstractMatrix)

Compute the matrix hyperbolic tangent of a square matrix A.

Base.Math.sechMethod
sech(A::AbstractMatrix)

Compute the matrix hyperbolic secant of square matrix A.

Base.Math.cschMethod
csch(A::AbstractMatrix)

Compute the matrix hyperbolic cosecant of square matrix A.

Base.Math.cothMethod
coth(A::AbstractMatrix)

Compute the matrix hyperbolic cotangent of square matrix A.

Base.acosMethod
acos(A::AbstractMatrix)

Compute the inverse matrix cosine of a square matrix A.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the inverse cosine. Otherwise, the inverse cosine is determined by using log and sqrt. For the theory and logarithmic formulas used to compute this function, see [AH16_1].

Examples

julia> acos(cos([0.5 0.1; -0.2 0.3]))
2×2 Array{Complex{Float64},2}:
0.5-8.32667e-17im  0.1+0.0im
-0.2+2.63678e-16im  0.3-3.46945e-16im
Base.asinMethod
asin(A::AbstractMatrix)

Compute the inverse matrix sine of a square matrix A.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the inverse sine. Otherwise, the inverse sine is determined by using log and sqrt. For the theory and logarithmic formulas used to compute this function, see [AH16_2].

Examples

julia> asin(sin([0.5 0.1; -0.2 0.3]))
2×2 Array{Complex{Float64},2}:
0.5-4.16334e-17im  0.1-5.55112e-17im
-0.2+9.71445e-17im  0.3-1.249e-16im
Base.atanMethod
atan(A::AbstractMatrix)

Compute the inverse matrix tangent of a square matrix A.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the inverse tangent. Otherwise, the inverse tangent is determined by using log. For the theory and logarithmic formulas used to compute this function, see [AH16_3].

Examples

julia> atan(tan([0.5 0.1; -0.2 0.3]))
2×2 Array{Complex{Float64},2}:
0.5+1.38778e-17im  0.1-2.77556e-17im
-0.2+6.93889e-17im  0.3-4.16334e-17im
Base.Math.asecMethod
asec(A::AbstractMatrix)

Compute the inverse matrix secant of A.

Base.Math.acscMethod
acsc(A::AbstractMatrix)

Compute the inverse matrix cosecant of A.

Base.Math.acotMethod
acot(A::AbstractMatrix)

Compute the inverse matrix cotangent of A.

Base.acoshMethod
acosh(A::AbstractMatrix)

Compute the inverse hyperbolic matrix cosine of a square matrix A. For the theory and logarithmic formulas used to compute this function, see [AH16_4].

Base.asinhMethod
asinh(A::AbstractMatrix)

Compute the inverse hyperbolic matrix sine of a square matrix A. For the theory and logarithmic formulas used to compute this function, see [AH16_5].

Base.atanhMethod
atanh(A::AbstractMatrix)

Compute the inverse hyperbolic matrix tangent of a square matrix A. For the theory and logarithmic formulas used to compute this function, see [AH16_6].

Base.Math.asechMethod
asech(A::AbstractMatrix)

Compute the inverse matrix hyperbolic secant of A.

Base.Math.acschMethod
acsch(A::AbstractMatrix)

Compute the inverse matrix hyperbolic cosecant of A.

Base.Math.acothMethod
acoth(A::AbstractMatrix)

Compute the inverse matrix hyperbolic cotangent of A.

LinearAlgebra.lyapFunction
lyap(A, C)

Computes the solution X to the continuous Lyapunov equation AX + XA' + C = 0, where no eigenvalue of A has a zero real part and no two eigenvalues are negative complex conjugates of each other.

Examples

julia> A = [3. 4.; 5. 6]
2×2 Array{Float64,2}:
3.0  4.0
5.0  6.0

julia> B = [1. 1.; 1. 2.]
2×2 Array{Float64,2}:
1.0  1.0
1.0  2.0

julia> X = lyap(A, B)
2×2 Array{Float64,2}:
0.5  -0.5
-0.5   0.25

julia> A*X + X*A' + B
2×2 Array{Float64,2}:
0.0          6.66134e-16
6.66134e-16  8.88178e-16
LinearAlgebra.sylvesterFunction
sylvester(A, B, C)

Computes the solution X to the Sylvester equation AX + XB + C = 0, where A, B and C have compatible dimensions and A and -B have no eigenvalues with equal real part.

Examples

julia> A = [3. 4.; 5. 6]
2×2 Array{Float64,2}:
3.0  4.0
5.0  6.0

julia> B = [1. 1.; 1. 2.]
2×2 Array{Float64,2}:
1.0  1.0
1.0  2.0

julia> C = [1. 2.; -2. 1]
2×2 Array{Float64,2}:
1.0  2.0
-2.0  1.0

julia> X = sylvester(A, B, C)
2×2 Array{Float64,2}:
-4.46667   1.93333
3.73333  -1.8

julia> A*X + X*B + C
2×2 Array{Float64,2}:
2.66454e-15  1.77636e-15
-3.77476e-15  4.44089e-16
LinearAlgebra.issuccessFunction
issuccess(F::Factorization)

Test that a factorization of a matrix succeeded.

julia> F = cholesky([1 0; 0 1]);

julia> LinearAlgebra.issuccess(F)
true

julia> F = lu([1 0; 0 0]; check = false);

julia> LinearAlgebra.issuccess(F)
false
LinearAlgebra.issymmetricFunction
issymmetric(A) -> Bool

Test whether a matrix is symmetric.

Examples

julia> a = [1 2; 2 -1]
2×2 Array{Int64,2}:
1   2
2  -1

julia> issymmetric(a)
true

julia> b = [1 im; -im 1]
2×2 Array{Complex{Int64},2}:
1+0im  0+1im
0-1im  1+0im

julia> issymmetric(b)
false
LinearAlgebra.isposdefFunction
isposdef(A) -> Bool

Test whether a matrix is positive definite (and Hermitian) by trying to perform a Cholesky factorization of A. See also isposdef!

Examples

julia> A = [1 2; 2 50]
2×2 Array{Int64,2}:
1   2
2  50

julia> isposdef(A)
true
LinearAlgebra.isposdef!Function
isposdef!(A) -> Bool

Test whether a matrix is positive definite (and Hermitian) by trying to perform a Cholesky factorization of A, overwriting A in the process. See also isposdef.

Examples

julia> A = [1. 2.; 2. 50.];

julia> isposdef!(A)
true

julia> A
2×2 Array{Float64,2}:
1.0  2.0
2.0  6.78233
LinearAlgebra.istrilFunction
istril(A::AbstractMatrix, k::Integer = 0) -> Bool

Test whether A is lower triangular starting from the kth superdiagonal.

Examples

julia> a = [1 2; 2 -1]
2×2 Array{Int64,2}:
1   2
2  -1

julia> istril(a)
false

julia> istril(a, 1)
true

julia> b = [1 0; -im -1]
2×2 Array{Complex{Int64},2}:
1+0im   0+0im
0-1im  -1+0im

julia> istril(b)
true

julia> istril(b, -1)
false
LinearAlgebra.istriuFunction
istriu(A::AbstractMatrix, k::Integer = 0) -> Bool

Test whether A is upper triangular starting from the kth superdiagonal.

Examples

julia> a = [1 2; 2 -1]
2×2 Array{Int64,2}:
1   2
2  -1

julia> istriu(a)
false

julia> istriu(a, -1)
true

julia> b = [1 im; 0 -1]
2×2 Array{Complex{Int64},2}:
1+0im   0+1im
0+0im  -1+0im

julia> istriu(b)
true

julia> istriu(b, 1)
false
LinearAlgebra.isdiagFunction
isdiag(A) -> Bool

Test whether a matrix is diagonal.

Examples

julia> a = [1 2; 2 -1]
2×2 Array{Int64,2}:
1   2
2  -1

julia> isdiag(a)
false

julia> b = [im 0; 0 -im]
2×2 Array{Complex{Int64},2}:
0+1im  0+0im
0+0im  0-1im

julia> isdiag(b)
true
LinearAlgebra.ishermitianFunction
ishermitian(A) -> Bool

Test whether a matrix is Hermitian.

Examples

julia> a = [1 2; 2 -1]
2×2 Array{Int64,2}:
1   2
2  -1

julia> ishermitian(a)
true

julia> b = [1 im; -im 1]
2×2 Array{Complex{Int64},2}:
1+0im  0+1im
0-1im  1+0im

julia> ishermitian(b)
true
Base.transposeFunction
transpose(A)

Lazy transpose. Mutating the returned object should appropriately mutate A. Often, but not always, yields Transpose(A), where Transpose is a lazy transpose wrapper. Note that this operation is recursive.

This operation is intended for linear algebra usage - for general data manipulation see permutedims, which is non-recursive.

Examples

julia> A = [3+2im 9+2im; 8+7im  4+6im]
2×2 Array{Complex{Int64},2}:
3+2im  9+2im
8+7im  4+6im

julia> transpose(A)
2×2 Transpose{Complex{Int64},Array{Complex{Int64},2}}:
3+2im  8+7im
9+2im  4+6im
LinearAlgebra.transpose!Function
transpose!(dest,src)

Transpose array src and store the result in the preallocated array dest, which should have a size corresponding to (size(src,2),size(src,1)). No in-place transposition is supported and unexpected results will happen if src and dest have overlapping memory regions.

Examples

julia> A = [3+2im 9+2im; 8+7im  4+6im]
2×2 Array{Complex{Int64},2}:
3+2im  9+2im
8+7im  4+6im

julia> B = zeros(Complex{Int64}, 2, 2)
2×2 Array{Complex{Int64},2}:
0+0im  0+0im
0+0im  0+0im

julia> transpose!(B, A);

julia> B
2×2 Array{Complex{Int64},2}:
3+2im  8+7im
9+2im  4+6im

julia> A
2×2 Array{Complex{Int64},2}:
3+2im  9+2im
8+7im  4+6im
Base.adjointFunction
adjoint(A)

Lazy adjoint (conjugate transposition) (also postfix '). Note that adjoint is applied recursively to elements.

This operation is intended for linear algebra usage - for general data manipulation see permutedims.

Examples

julia> A = [3+2im 9+2im; 8+7im  4+6im]
2×2 Array{Complex{Int64},2}:
3+2im  9+2im
8+7im  4+6im

3-2im  8-7im
9-2im  4-6im
LinearAlgebra.adjoint!Function
adjoint!(dest,src)

Conjugate transpose array src and store the result in the preallocated array dest, which should have a size corresponding to (size(src,2),size(src,1)). No in-place transposition is supported and unexpected results will happen if src and dest have overlapping memory regions.

Examples

julia> A = [3+2im 9+2im; 8+7im  4+6im]
2×2 Array{Complex{Int64},2}:
3+2im  9+2im
8+7im  4+6im

julia> B = zeros(Complex{Int64}, 2, 2)
2×2 Array{Complex{Int64},2}:
0+0im  0+0im
0+0im  0+0im

julia> B
2×2 Array{Complex{Int64},2}:
3-2im  8-7im
9-2im  4-6im

julia> A
2×2 Array{Complex{Int64},2}:
3+2im  9+2im
8+7im  4+6im
Base.copyMethod
copy(A::Transpose)
copy(A::Adjoint)

Eagerly evaluate the lazy matrix transpose/adjoint. Note that the transposition is applied recursively to elements.

This operation is intended for linear algebra usage - for general data manipulation see permutedims, which is non-recursive.

Examples

julia> A = [1 2im; -3im 4]
2×2 Array{Complex{Int64},2}:
1+0im  0+2im
0-3im  4+0im

julia> T = transpose(A)
2×2 Transpose{Complex{Int64},Array{Complex{Int64},2}}:
1+0im  0-3im
0+2im  4+0im

julia> copy(T)
2×2 Array{Complex{Int64},2}:
1+0im  0-3im
0+2im  4+0im
LinearAlgebra.stride1Function
stride1(A) -> Int

Return the distance between successive array elements in dimension 1 in units of element size.

Examples

julia> A = [1,2,3,4]
4-element Array{Int64,1}:
1
2
3
4

julia> LinearAlgebra.stride1(A)
1

julia> B = view(A, 2:2:4)
2-element view(::Array{Int64,1}, 2:2:4) with eltype Int64:
2
4

julia> LinearAlgebra.stride1(B)
2
LinearAlgebra.checksquareFunction
LinearAlgebra.checksquare(A)

Check that a matrix is square, then return its common dimension. For multiple arguments, return a vector.

Examples

julia> A = fill(1, (4,4)); B = fill(1, (5,5));

julia> LinearAlgebra.checksquare(A, B)
2-element Array{Int64,1}:
4
5

## 底层矩阵运算

LinearAlgebra.mul!Function
mul!(Y, A, B) -> Y

Calculates the matrix-matrix or matrix-vector product $AB$ and stores the result in Y, overwriting the existing value of Y. Note that Y must not be aliased with either A or B.

Examples

julia> A=[1.0 2.0; 3.0 4.0]; B=[1.0 1.0; 1.0 1.0]; Y = similar(B); mul!(Y, A, B);

julia> Y
2×2 Array{Float64,2}:
3.0  3.0
7.0  7.0

Implementation

For custom matrix and vector types, it is recommended to implement 5-argument mul! rather than implementing 3-argument mul! directly if possible.

mul!(C, A, B, α, β) -> C

Combined inplace matrix-matrix or matrix-vector multiply-add $A B α + C β$. The result is stored in C by overwriting it. Note that C must not be aliased with either A or B.

Julia 1.3

Five-argument mul! requires at least Julia 1.3.

Examples

julia> A=[1.0 2.0; 3.0 4.0]; B=[1.0 1.0; 1.0 1.0]; C=[1.0 2.0; 3.0 4.0];

julia> mul!(C, A, B, 100.0, 10.0) === C
true

julia> C
2×2 Array{Float64,2}:
310.0  320.0
730.0  740.0
LinearAlgebra.lmul!Function
lmul!(a::Number, B::AbstractArray)

Scale an array B by a scalar a overwriting B in-place. Use rmul! to multiply scalar from right. The scaling operation respects the semantics of the multiplication * between a and an element of B. In particular, this also applies to multiplication involving non-finite numbers such as NaN and ±Inf.

Julia 1.1

Prior to Julia 1.1, NaN and ±Inf entries in B were treated inconsistently.

Examples

julia> B = [1 2; 3 4]
2×2 Array{Int64,2}:
1  2
3  4

julia> lmul!(2, B)
2×2 Array{Int64,2}:
2  4
6  8

julia> lmul!(0.0, [Inf])
1-element Array{Float64,1}:
NaN
lmul!(A, B)

Calculate the matrix-matrix product $AB$, overwriting B, and return the result. Here, A must be of special matrix type, like, e.g., Diagonal, UpperTriangular or LowerTriangular, or of some orthogonal type, see QR.

Examples

julia> B = [0 1; 1 0];

julia> A = LinearAlgebra.UpperTriangular([1 2; 0 3]);

julia> LinearAlgebra.lmul!(A, B);

julia> B
2×2 Array{Int64,2}:
2  1
3  0

julia> B = [1.0 2.0; 3.0 4.0];

julia> F = qr([0 1; -1 0]);

julia> lmul!(F.Q, B)
2×2 Array{Float64,2}:
3.0  4.0
1.0  2.0
LinearAlgebra.rmul!Function
rmul!(A::AbstractArray, b::Number)

Scale an array A by a scalar b overwriting A in-place. Use lmul! to multiply scalar from left. The scaling operation respects the semantics of the multiplication * between an element of A and b. In particular, this also applies to multiplication involving non-finite numbers such as NaN and ±Inf.

Julia 1.1

Prior to Julia 1.1, NaN and ±Inf entries in A were treated inconsistently.

Examples

julia> A = [1 2; 3 4]
2×2 Array{Int64,2}:
1  2
3  4

julia> rmul!(A, 2)
2×2 Array{Int64,2}:
2  4
6  8

julia> rmul!([NaN], 0.0)
1-element Array{Float64,1}:
NaN
rmul!(A, B)

Calculate the matrix-matrix product $AB$, overwriting A, and return the result. Here, B must be of special matrix type, like, e.g., Diagonal, UpperTriangular or LowerTriangular, or of some orthogonal type, see QR.

Examples

julia> A = [0 1; 1 0];

julia> B = LinearAlgebra.UpperTriangular([1 2; 0 3]);

julia> LinearAlgebra.rmul!(A, B);

julia> A
2×2 Array{Int64,2}:
0  3
1  2

julia> A = [1.0 2.0; 3.0 4.0];

julia> F = qr([0 1; -1 0]);

julia> rmul!(A, F.Q)
2×2 Array{Float64,2}:
2.0  1.0
4.0  3.0
LinearAlgebra.ldiv!Function
ldiv!(Y, A, B) -> Y

Compute A \ B in-place and store the result in Y, returning the result.

The argument A should not be a matrix. Rather, instead of matrices it should be a factorization object (e.g. produced by factorize or cholesky). The reason for this is that factorization itself is both expensive and typically allocates memory (although it can also be done in-place via, e.g., lu!), and performance-critical situations requiring ldiv! usually also require fine-grained control over the factorization of A.

Examples

julia> A = [1 2.2 4; 3.1 0.2 3; 4 1 2];

julia> X = [1; 2.5; 3];

julia> Y = zero(X);

julia> ldiv!(Y, qr(A), X);

julia> Y
3-element Array{Float64,1}:
0.7128099173553719
-0.051652892561983674
0.10020661157024757

julia> A\X
3-element Array{Float64,1}:
0.7128099173553719
-0.05165289256198333
0.10020661157024785
ldiv!(A, B)

Compute A \ B in-place and overwriting B to store the result.

The argument A should not be a matrix. Rather, instead of matrices it should be a factorization object (e.g. produced by factorize or cholesky). The reason for this is that factorization itself is both expensive and typically allocates memory (although it can also be done in-place via, e.g., lu!), and performance-critical situations requiring ldiv! usually also require fine-grained control over the factorization of A.

Examples

julia> A = [1 2.2 4; 3.1 0.2 3; 4 1 2];

julia> X = [1; 2.5; 3];

julia> Y = copy(X);

julia> ldiv!(qr(A), X);

julia> X
3-element Array{Float64,1}:
0.7128099173553719
-0.051652892561983674
0.10020661157024757

julia> A\Y
3-element Array{Float64,1}:
0.7128099173553719
-0.05165289256198333
0.10020661157024785
ldiv!(a::Number, B::AbstractArray)

Divide each entry in an array B by a scalar a overwriting B in-place. Use rdiv! to divide scalar from right.

Examples

julia> B = [1.0 2.0; 3.0 4.0]
2×2 Array{Float64,2}:
1.0  2.0
3.0  4.0

julia> ldiv!(2.0, B)
2×2 Array{Float64,2}:
0.5  1.0
1.5  2.0
LinearAlgebra.rdiv!Function
rdiv!(A, B)

Compute A / B in-place and overwriting A to store the result.

The argument B should not be a matrix. Rather, instead of matrices it should be a factorization object (e.g. produced by factorize or cholesky). The reason for this is that factorization itself is both expensive and typically allocates memory (although it can also be done in-place via, e.g., lu!), and performance-critical situations requiring rdiv! usually also require fine-grained control over the factorization of B.

rdiv!(A::AbstractArray, b::Number)

Divide each entry in an array A by a scalar b overwriting A in-place. Use ldiv! to divide scalar from left.

Examples

julia> A = [1.0 2.0; 3.0 4.0]
2×2 Array{Float64,2}:
1.0  2.0
3.0  4.0

julia> rdiv!(A, 2.0)
2×2 Array{Float64,2}:
0.5  1.0
1.5  2.0

## BLAS 函数

LinearAlgebra.BLAS 提供了一些 BLAS 函数的封装。那些改写了某个输入数组的 BLAS 函数的名称以 '!' 结尾。通常，一个 BLAS 函数定义了四个方法，分别针对 Float64Float32ComplexF64ComplexF32 数组。

### BLAS 字符参数

#### 乘法顺序

side含义
'L'参数位于矩阵与矩阵运算的边。
'R'参数位于矩阵与矩阵运算的边。

#### 三角引用

uplo/ul含义
'U'只会使用矩阵的三角部分。
'L'只会使用矩阵的三角部分。

#### 转置运算

trans/tX含义
'N'输入矩阵 X 不被转置或共轭。
'T'输入矩阵 X 会被转置。
'C'输入矩阵 X 会被共轭转置。

#### 单位对角线

diag/dX含义
'N'矩阵 X 对角线上的值会被读取。
'U'矩阵 X 对角线上假设全为一。
LinearAlgebra.BLAS.dotuFunction
dotu(n, X, incx, Y, incy)

Dot function for two complex vectors consisting of n elements of array X with stride incx and n elements of array Y with stride incy.

Examples

julia> BLAS.dotu(10, fill(1.0im, 10), 1, fill(1.0+im, 20), 2)
-10.0 + 10.0im
LinearAlgebra.BLAS.dotcFunction
dotc(n, X, incx, U, incy)

Dot function for two complex vectors, consisting of n elements of array X with stride incx and n elements of array U with stride incy, conjugating the first vector.

Examples

julia> BLAS.dotc(10, fill(1.0im, 10), 1, fill(1.0+im, 20), 2)
10.0 - 10.0im
LinearAlgebra.BLAS.blascopy!Function
blascopy!(n, X, incx, Y, incy)

Copy n elements of array X with stride incx to array Y with stride incy. Returns Y.

LinearAlgebra.BLAS.nrm2Function
nrm2(n, X, incx)

2-norm of a vector consisting of n elements of array X with stride incx.

Examples

julia> BLAS.nrm2(4, fill(1.0, 8), 2)
2.0

julia> BLAS.nrm2(1, fill(1.0, 8), 2)
1.0
LinearAlgebra.BLAS.asumFunction
asum(n, X, incx)

Sum of the absolute values of the first n elements of array X with stride incx.

Examples

julia> BLAS.asum(5, fill(1.0im, 10), 2)
5.0

julia> BLAS.asum(2, fill(1.0im, 10), 5)
2.0
LinearAlgebra.axpy!Function
axpy!(a, X, Y)

Overwrite Y with a*X + Y, where a is a scalar. Return Y.

Examples

julia> x = [1; 2; 3];

julia> y = [4; 5; 6];

julia> BLAS.axpy!(2, x, y)
3-element Array{Int64,1}:
6
9
12
LinearAlgebra.BLAS.scal!Function
scal!(n, a, X, incx)

Overwrite X with a*X for the first n elements of array X with stride incx. Returns X.

LinearAlgebra.BLAS.scalFunction
scal(n, a, X, incx)

Return X scaled by a for the first n elements of array X with stride incx.

LinearAlgebra.BLAS.ger!Function
ger!(alpha, x, y, A)

Rank-1 update of the matrix A with vectors x and y as alpha*x*y' + A.

LinearAlgebra.BLAS.syr!Function
syr!(uplo, alpha, x, A)

Rank-1 update of the symmetric matrix A with vector x as alpha*x*transpose(x) + A. uplo controls which triangle of A is updated. Returns A.

LinearAlgebra.BLAS.syrk!Function
syrk!(uplo, trans, alpha, A, beta, C)

Rank-k update of the symmetric matrix C as alpha*A*transpose(A) + beta*C or alpha*transpose(A)*A + beta*C according to trans. Only the uplo triangle of C is used. Returns C.

LinearAlgebra.BLAS.her!Function
her!(uplo, alpha, x, A)

Methods for complex arrays only. Rank-1 update of the Hermitian matrix A with vector x as alpha*x*x' + A. uplo controls which triangle of A is updated. Returns A.

LinearAlgebra.BLAS.herk!Function
herk!(uplo, trans, alpha, A, beta, C)

Methods for complex arrays only. Rank-k update of the Hermitian matrix C as alpha*A*A' + beta*C or alpha*A'*A + beta*C according to trans. Only the uplo triangle of C is updated. Returns C.

LinearAlgebra.BLAS.gbmv!Function
gbmv!(trans, m, kl, ku, alpha, A, x, beta, y)

Update vector y as alpha*A*x + beta*y or alpha*A'*x + beta*y according to trans. The matrix A is a general band matrix of dimension m by size(A,2) with kl sub-diagonals and ku super-diagonals. alpha and beta are scalars. Return the updated y.

LinearAlgebra.BLAS.gbmvFunction
gbmv(trans, m, kl, ku, alpha, A, x)

Return alpha*A*x or alpha*A'*x according to trans. The matrix A is a general band matrix of dimension m by size(A,2) with kl sub-diagonals and ku super-diagonals, and alpha is a scalar.

LinearAlgebra.BLAS.sbmv!Function
sbmv!(uplo, k, alpha, A, x, beta, y)

Update vector y as alpha*A*x + beta*y where A is a a symmetric band matrix of order size(A,2) with k super-diagonals stored in the argument A. The storage layout for A is described the reference BLAS module, level-2 BLAS at http://www.netlib.org/lapack/explore-html/. Only the uplo triangle of A is used.

Return the updated y.

LinearAlgebra.BLAS.sbmvMethod
sbmv(uplo, k, alpha, A, x)

Return alpha*A*x where A is a symmetric band matrix of order size(A,2) with k super-diagonals stored in the argument A. Only the uplo triangle of A is used.

LinearAlgebra.BLAS.sbmvMethod
sbmv(uplo, k, A, x)

Return A*x where A is a symmetric band matrix of order size(A,2) with k super-diagonals stored in the argument A. Only the uplo triangle of A is used.

LinearAlgebra.BLAS.gemm!Function
gemm!(tA, tB, alpha, A, B, beta, C)

Update C as alpha*A*B + beta*C or the other three variants according to tA and tB. Return the updated C.

LinearAlgebra.BLAS.gemv!Function
gemv!(tA, alpha, A, x, beta, y)

Update the vector y as alpha*A*x + beta*y or alpha*A'x + beta*y according to tA. alpha and beta are scalars. Return the updated y.

LinearAlgebra.BLAS.symm!Function
symm!(side, ul, alpha, A, B, beta, C)

Update C as alpha*A*B + beta*C or alpha*B*A + beta*C according to side. A is assumed to be symmetric. Only the ul triangle of A is used. Return the updated C.

LinearAlgebra.BLAS.symv!Function
symv!(ul, alpha, A, x, beta, y)

Update the vector y as alpha*A*x + beta*y. A is assumed to be symmetric. Only the ul triangle of A is used. alpha and beta are scalars. Return the updated y.

LinearAlgebra.BLAS.symvMethod
symv(ul, alpha, A, x)

Return alpha*A*x. A is assumed to be symmetric. Only the ul triangle of A is used. alpha is a scalar.

LinearAlgebra.IConstant
I

An object of type UniformScaling, representing an identity matrix of any size.

Examples

julia> fill(1, (5,6)) * I == fill(1, (5,6))
true

julia> [1 2im 3; 1im 2 3] * I
2×3 Array{Complex{Int64},2}:
1+0im  0+2im  3+0im
0+1im  2+0im  3+0im

## LAPACK 函数

LinearAlgebra.LAPACK 提供了一些针对线性代数的 LAPACK 函数的封装。那些改写了输入数组的函数的名称以 '!' 结尾。

LinearAlgebra.LAPACK.gbtrf!Function
gbtrf!(kl, ku, m, AB) -> (AB, ipiv)

Compute the LU factorization of a banded matrix AB. kl is the first subdiagonal containing a nonzero band, ku is the last superdiagonal containing one, and m is the first dimension of the matrix AB. Returns the LU factorization in-place and ipiv, the vector of pivots used.

LinearAlgebra.LAPACK.gbtrs!Function
gbtrs!(trans, kl, ku, m, AB, ipiv, B)

Solve the equation AB * X = B. trans determines the orientation of AB. It may be N (no transpose), T (transpose), or C (conjugate transpose). kl is the first subdiagonal containing a nonzero band, ku is the last superdiagonal containing one, and m is the first dimension of the matrix AB. ipiv is the vector of pivots returned from gbtrf!. Returns the vector or matrix X, overwriting B in-place.

LinearAlgebra.LAPACK.gebal!Function
gebal!(job, A) -> (ilo, ihi, scale)

Balance the matrix A before computing its eigensystem or Schur factorization. job can be one of N (A will not be permuted or scaled), P (A will only be permuted), S (A will only be scaled), or B (A will be both permuted and scaled). Modifies A in-place and returns ilo, ihi, and scale. If permuting was turned on, A[i,j] = 0 if j > i and 1 < j < ilo or j > ihi. scale contains information about the scaling/permutations performed.

LinearAlgebra.LAPACK.gebak!Function
gebak!(job, side, ilo, ihi, scale, V)

Transform the eigenvectors V of a matrix balanced using gebal! to the unscaled/unpermuted eigenvectors of the original matrix. Modifies V in-place. side can be L (left eigenvectors are transformed) or R (right eigenvectors are transformed).

LinearAlgebra.LAPACK.gebrd!Function
gebrd!(A) -> (A, d, e, tauq, taup)

Reduce A in-place to bidiagonal form A = QBP'. Returns A, containing the bidiagonal matrix B; d, containing the diagonal elements of B; e, containing the off-diagonal elements of B; tauq, containing the elementary reflectors representing Q; and taup, containing the elementary reflectors representing P.

LinearAlgebra.LAPACK.gelqf!Function
gelqf!(A, tau)

Compute the LQ factorization of A, A = LQ. tau contains scalars which parameterize the elementary reflectors of the factorization. tau must have length greater than or equal to the smallest dimension of A.

Returns A and tau modified in-place.

gelqf!(A) -> (A, tau)

Compute the LQ factorization of A, A = LQ.

Returns A, modified in-place, and tau, which contains scalars which parameterize the elementary reflectors of the factorization.

LinearAlgebra.LAPACK.geqlf!Function
geqlf!(A, tau)

Compute the QL factorization of A, A = QL. tau contains scalars which parameterize the elementary reflectors of the factorization. tau must have length greater than or equal to the smallest dimension of A.

Returns A and tau modified in-place.

geqlf!(A) -> (A, tau)

Compute the QL factorization of A, A = QL.

Returns A, modified in-place, and tau, which contains scalars which parameterize the elementary reflectors of the factorization.

LinearAlgebra.LAPACK.geqrf!Function
geqrf!(A, tau)

Compute the QR factorization of A, A = QR. tau contains scalars which parameterize the elementary reflectors of the factorization. tau must have length greater than or equal to the smallest dimension of A.

Returns A and tau modified in-place.

geqrf!(A) -> (A, tau)

Compute the QR factorization of A, A = QR.

Returns A, modified in-place, and tau, which contains scalars which parameterize the elementary reflectors of the factorization.

LinearAlgebra.LAPACK.geqp3!Function
geqp3!(A, jpvt, tau)

Compute the pivoted QR factorization of A, AP = QR using BLAS level 3. P is a pivoting matrix, represented by jpvt. tau stores the elementary reflectors. jpvt must have length length greater than or equal to n if A is an (m x n) matrix. tau must have length greater than or equal to the smallest dimension of A.

A, jpvt, and tau are modified in-place.

geqp3!(A, jpvt) -> (A, jpvt, tau)

Compute the pivoted QR factorization of A, AP = QR using BLAS level 3. P is a pivoting matrix, represented by jpvt. jpvt must have length greater than or equal to n if A is an (m x n) matrix.

Returns A and jpvt, modified in-place, and tau, which stores the elementary reflectors.

geqp3!(A) -> (A, jpvt, tau)

Compute the pivoted QR factorization of A, AP = QR using BLAS level 3.

Returns A, modified in-place, jpvt, which represents the pivoting matrix P, and tau, which stores the elementary reflectors.

LinearAlgebra.LAPACK.gerqf!Function
gerqf!(A, tau)

Compute the RQ factorization of A, A = RQ. tau contains scalars which parameterize the elementary reflectors of the factorization. tau must have length greater than or equal to the smallest dimension of A.

Returns A and tau modified in-place.

gerqf!(A) -> (A, tau)

Compute the RQ factorization of A, A = RQ.

Returns A, modified in-place, and tau, which contains scalars which parameterize the elementary reflectors of the factorization.

LinearAlgebra.LAPACK.geqrt!Function
geqrt!(A, T)

Compute the blocked QR factorization of A, A = QR. T contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization. The first dimension of T sets the block size and it must be between 1 and n. The second dimension of T must equal the smallest dimension of A.

Returns A and T modified in-place.

geqrt!(A, nb) -> (A, T)

Compute the blocked QR factorization of A, A = QR. nb sets the block size and it must be between 1 and n, the second dimension of A.

Returns A, modified in-place, and T, which contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization.

LinearAlgebra.LAPACK.geqrt3!Function
geqrt3!(A, T)

Recursively computes the blocked QR factorization of A, A = QR. T contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization. The first dimension of T sets the block size and it must be between 1 and n. The second dimension of T must equal the smallest dimension of A.

Returns A and T modified in-place.

geqrt3!(A) -> (A, T)

Recursively computes the blocked QR factorization of A, A = QR.

Returns A, modified in-place, and T, which contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization.

LinearAlgebra.LAPACK.getrf!Function
getrf!(A) -> (A, ipiv, info)

Compute the pivoted LU factorization of A, A = LU.

Returns A, modified in-place, ipiv, the pivoting information, and an info code which indicates success (info = 0), a singular value in U (info = i, in which case U[i,i] is singular), or an error code (info < 0).

LinearAlgebra.LAPACK.tzrzf!Function
tzrzf!(A) -> (A, tau)

Transforms the upper trapezoidal matrix A to upper triangular form in-place. Returns A and tau, the scalar parameters for the elementary reflectors of the transformation.

LinearAlgebra.LAPACK.ormrz!Function
ormrz!(side, trans, A, tau, C)

Multiplies the matrix C by Q from the transformation supplied by tzrzf!. Depending on side or trans the multiplication can be left-sided (side = L, Q*C) or right-sided (side = R, C*Q) and Q can be unmodified (trans = N), transposed (trans = T), or conjugate transposed (trans = C). Returns matrix C which is modified in-place with the result of the multiplication.

LinearAlgebra.LAPACK.gels!Function
gels!(trans, A, B) -> (F, B, ssr)

Solves the linear equation A * X = B, transpose(A) * X = B, or adjoint(A) * X = B using a QR or LQ factorization. Modifies the matrix/vector B in place with the solution. A is overwritten with its QR or LQ factorization. trans may be one of N (no modification), T (transpose), or C (conjugate transpose). gels! searches for the minimum norm/least squares solution. A may be under or over determined. The solution is returned in B.

LinearAlgebra.LAPACK.gesv!Function
gesv!(A, B) -> (B, A, ipiv)

Solves the linear equation A * X = B where A is a square matrix using the LU factorization of A. A is overwritten with its LU factorization and B is overwritten with the solution X. ipiv contains the pivoting information for the LU factorization of A.

LinearAlgebra.LAPACK.getrs!Function
getrs!(trans, A, ipiv, B)

Solves the linear equation A * X = B, transpose(A) * X = B, or adjoint(A) * X = B for square A. Modifies the matrix/vector B in place with the solution. A is the LU factorization from getrf!, with ipiv the pivoting information. trans may be one of N (no modification), T (transpose), or C (conjugate transpose).

LinearAlgebra.LAPACK.getri!Function
getri!(A, ipiv)

Computes the inverse of A, using its LU factorization found by getrf!. ipiv is the pivot information output and A contains the LU factorization of getrf!. A is overwritten with its inverse.

LinearAlgebra.LAPACK.gesvx!Function
gesvx!(fact, trans, A, AF, ipiv, equed, R, C, B) -> (X, equed, R, C, B, rcond, ferr, berr, work)

Solves the linear equation A * X = B (trans = N), transpose(A) * X = B (trans = T), or adjoint(A) * X = B (trans = C) using the LU factorization of A. fact may be E, in which case A will be equilibrated and copied to AF; F, in which case AF and ipiv from a previous LU factorization are inputs; or N, in which case A will be copied to AF and then factored. If fact = F, equed may be N, meaning A has not been equilibrated; R, meaning A was multiplied by Diagonal(R) from the left; C, meaning A was multiplied by Diagonal(C) from the right; or B, meaning A was multiplied by Diagonal(R) from the left and Diagonal(C) from the right. If fact = F and equed = R or B the elements of R must all be positive. If fact = F and equed = C or B the elements of C must all be positive.

Returns the solution X; equed, which is an output if fact is not N, and describes the equilibration that was performed; R, the row equilibration diagonal; C, the column equilibration diagonal; B, which may be overwritten with its equilibrated form Diagonal(R)*B (if trans = N and equed = R,B) or Diagonal(C)*B (if trans = T,C and equed = C,B); rcond, the reciprocal condition number of A after equilbrating; ferr, the forward error bound for each solution vector in X; berr, the forward error bound for each solution vector in X; and work, the reciprocal pivot growth factor.

gesvx!(A, B)

The no-equilibration, no-transpose simplification of gesvx!.

LinearAlgebra.LAPACK.gelsd!Function
gelsd!(A, B, rcond) -> (B, rnk)

Computes the least norm solution of A * X = B by finding the SVD factorization of A, then dividing-and-conquering the problem. B is overwritten with the solution X. Singular values below rcond will be treated as zero. Returns the solution in B and the effective rank of A in rnk.

LinearAlgebra.LAPACK.gelsy!Function
gelsy!(A, B, rcond) -> (B, rnk)

Computes the least norm solution of A * X = B by finding the full QR factorization of A, then dividing-and-conquering the problem. B is overwritten with the solution X. Singular values below rcond will be treated as zero. Returns the solution in B and the effective rank of A in rnk.

LinearAlgebra.LAPACK.gglse!Function
gglse!(A, c, B, d) -> (X,res)

Solves the equation A * x = c where x is subject to the equality constraint B * x = d. Uses the formula ||c - A*x||^2 = 0 to solve. Returns X and the residual sum-of-squares.

LinearAlgebra.LAPACK.geev!Function
geev!(jobvl, jobvr, A) -> (W, VL, VR)

Finds the eigensystem of A. If jobvl = N, the left eigenvectors of A aren't computed. If jobvr = N, the right eigenvectors of A aren't computed. If jobvl = V or jobvr = V, the corresponding eigenvectors are computed. Returns the eigenvalues in W, the right eigenvectors in VR, and the left eigenvectors in VL.

LinearAlgebra.LAPACK.gesdd!Function
gesdd!(job, A) -> (U, S, VT)

Finds the singular value decomposition of A, A = U * S * V', using a divide and conquer approach. If job = A, all the columns of U and the rows of V' are computed. If job = N, no columns of U or rows of V' are computed. If job = O, A is overwritten with the columns of (thin) U and the rows of (thin) V'. If job = S, the columns of (thin) U and the rows of (thin) V' are computed and returned separately.

LinearAlgebra.LAPACK.gesvd!Function
gesvd!(jobu, jobvt, A) -> (U, S, VT)

Finds the singular value decomposition of A, A = U * S * V'. If jobu = A, all the columns of U are computed. If jobvt = A all the rows of V' are computed. If jobu = N, no columns of U are computed. If jobvt = N no rows of V' are computed. If jobu = O, A is overwritten with the columns of (thin) U. If jobvt = O, A is overwritten with the rows of (thin) V'. If jobu = S, the columns of (thin) U are computed and returned separately. If jobvt = S the rows of (thin) V' are computed and returned separately. jobu and jobvt can't both be O.

Returns U, S, and Vt, where S are the singular values of A.

LinearAlgebra.LAPACK.ggsvd!Function
ggsvd!(jobu, jobv, jobq, A, B) -> (U, V, Q, alpha, beta, k, l, R)

Finds the generalized singular value decomposition of A and B, U'*A*Q = D1*R and V'*B*Q = D2*R. D1 has alpha on its diagonal and D2 has beta on its diagonal. If jobu = U, the orthogonal/unitary matrix U is computed. If jobv = V the orthogonal/unitary matrix V is computed. If jobq = Q, the orthogonal/unitary matrix Q is computed. If jobu, jobv or jobq is N, that matrix is not computed. This function is only available in LAPACK versions prior to 3.6.0.

LinearAlgebra.LAPACK.ggsvd3!Function
ggsvd3!(jobu, jobv, jobq, A, B) -> (U, V, Q, alpha, beta, k, l, R)

Finds the generalized singular value decomposition of A and B, U'*A*Q = D1*R and V'*B*Q = D2*R. D1 has alpha on its diagonal and D2 has beta on its diagonal. If jobu = U, the orthogonal/unitary matrix U is computed. If jobv = V the orthogonal/unitary matrix V is computed. If jobq = Q, the orthogonal/unitary matrix Q is computed. If jobu, jobv, or jobq is N, that matrix is not computed. This function requires LAPACK 3.6.0.

LinearAlgebra.LAPACK.geevx!Function
geevx!(balanc, jobvl, jobvr, sense, A) -> (A, w, VL, VR, ilo, ihi, scale, abnrm, rconde, rcondv)

Finds the eigensystem of A with matrix balancing. If jobvl = N, the left eigenvectors of A aren't computed. If jobvr = N, the right eigenvectors of A aren't computed. If jobvl = V or jobvr = V, the corresponding eigenvectors are computed. If balanc = N, no balancing is performed. If balanc = P, A is permuted but not scaled. If balanc = S, A is scaled but not permuted. If balanc = B, A is permuted and scaled. If sense = N, no reciprocal condition numbers are computed. If sense = E, reciprocal condition numbers are computed for the eigenvalues only. If sense = V, reciprocal condition numbers are computed for the right eigenvectors only. If sense = B, reciprocal condition numbers are computed for the right eigenvectors and the eigenvectors. If sense = E,B, the right and left eigenvectors must be computed.

LinearAlgebra.LAPACK.ggev!Function
ggev!(jobvl, jobvr, A, B) -> (alpha, beta, vl, vr)

Finds the generalized eigendecomposition of A and B. If jobvl = N, the left eigenvectors aren't computed. If jobvr = N, the right eigenvectors aren't computed. If jobvl = V or jobvr = V, the corresponding eigenvectors are computed.

LinearAlgebra.LAPACK.gtsv!Function
gtsv!(dl, d, du, B)

Solves the equation A * X = B where A is a tridiagonal matrix with dl on the subdiagonal, d on the diagonal, and du on the superdiagonal.

Overwrites B with the solution X and returns it.

LinearAlgebra.LAPACK.gttrf!Function
gttrf!(dl, d, du) -> (dl, d, du, du2, ipiv)

Finds the LU factorization of a tridiagonal matrix with dl on the subdiagonal, d on the diagonal, and du on the superdiagonal.

Modifies dl, d, and du in-place and returns them and the second superdiagonal du2 and the pivoting vector ipiv.

LinearAlgebra.LAPACK.gttrs!Function
gttrs!(trans, dl, d, du, du2, ipiv, B)

Solves the equation A * X = B (trans = N), transpose(A) * X = B (trans = T), or adjoint(A) * X = B (trans = C) using the LU factorization computed by gttrf!. B is overwritten with the solution X.

LinearAlgebra.LAPACK.orglq!Function
orglq!(A, tau, k = length(tau))

Explicitly finds the matrix Q of a LQ factorization after calling gelqf! on A. Uses the output of gelqf!. A is overwritten by Q.

LinearAlgebra.LAPACK.orgqr!Function
orgqr!(A, tau, k = length(tau))

Explicitly finds the matrix Q of a QR factorization after calling geqrf! on A. Uses the output of geqrf!. A is overwritten by Q.

LinearAlgebra.LAPACK.orgql!Function
orgql!(A, tau, k = length(tau))

Explicitly finds the matrix Q of a QL factorization after calling geqlf! on A. Uses the output of geqlf!. A is overwritten by Q.

LinearAlgebra.LAPACK.orgrq!Function
orgrq!(A, tau, k = length(tau))

Explicitly finds the matrix Q of a RQ factorization after calling gerqf! on A. Uses the output of gerqf!. A is overwritten by Q.

LinearAlgebra.LAPACK.ormlq!Function
ormlq!(side, trans, A, tau, C)

Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a LQ factorization of A computed using gelqf!. C is overwritten.

LinearAlgebra.LAPACK.ormqr!Function
ormqr!(side, trans, A, tau, C)

Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a QR factorization of A computed using geqrf!. C is overwritten.

LinearAlgebra.LAPACK.ormql!Function
ormql!(side, trans, A, tau, C)

Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a QL factorization of A computed using geqlf!. C is overwritten.

LinearAlgebra.LAPACK.ormrq!Function
ormrq!(side, trans, A, tau, C)

Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a RQ factorization of A computed using gerqf!. C is overwritten.

LinearAlgebra.LAPACK.gemqrt!Function
gemqrt!(side, trans, V, T, C)

Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a QR factorization of A computed using geqrt!. C is overwritten.

LinearAlgebra.LAPACK.posv!Function
posv!(uplo, A, B) -> (A, B)

Finds the solution to A * X = B where A is a symmetric or Hermitian positive definite matrix. If uplo = U the upper Cholesky decomposition of A is computed. If uplo = L the lower Cholesky decomposition of A is computed. A is overwritten by its Cholesky decomposition. B is overwritten with the solution X.

LinearAlgebra.LAPACK.potrf!Function
potrf!(uplo, A)

Computes the Cholesky (upper if uplo = U, lower if uplo = L) decomposition of positive-definite matrix A. A is overwritten and returned with an info code.

LinearAlgebra.LAPACK.potri!Function
potri!(uplo, A)

Computes the inverse of positive-definite matrix A after calling potrf! to find its (upper if uplo = U, lower if uplo = L) Cholesky decomposition.

A is overwritten by its inverse and returned.

LinearAlgebra.LAPACK.potrs!Function
potrs!(uplo, A, B)

Finds the solution to A * X = B where A is a symmetric or Hermitian positive definite matrix whose Cholesky decomposition was computed by potrf!. If uplo = U the upper Cholesky decomposition of A was computed. If uplo = L the lower Cholesky decomposition of A was computed. B is overwritten with the solution X.

LinearAlgebra.LAPACK.pstrf!Function
pstrf!(uplo, A, tol) -> (A, piv, rank, info)

Computes the (upper if uplo = U, lower if uplo = L) pivoted Cholesky decomposition of positive-definite matrix A with a user-set tolerance tol. A is overwritten by its Cholesky decomposition.

Returns A, the pivots piv, the rank of A, and an info code. If info = 0, the factorization succeeded. If info = i > 0, then A is indefinite or rank-deficient.

LinearAlgebra.LAPACK.ptsv!Function
ptsv!(D, E, B)

Solves A * X = B for positive-definite tridiagonal A. D is the diagonal of A and E is the off-diagonal. B is overwritten with the solution X and returned.

LinearAlgebra.LAPACK.pttrf!Function
pttrf!(D, E)

Computes the LDLt factorization of a positive-definite tridiagonal matrix with D as diagonal and E as off-diagonal. D and E are overwritten and returned.

LinearAlgebra.LAPACK.pttrs!Function
pttrs!(D, E, B)

Solves A * X = B for positive-definite tridiagonal A with diagonal D and off-diagonal E after computing A's LDLt factorization using pttrf!. B is overwritten with the solution X.

LinearAlgebra.LAPACK.trtri!Function
trtri!(uplo, diag, A)

Finds the inverse of (upper if uplo = U, lower if uplo = L) triangular matrix A. If diag = N, A has non-unit diagonal elements. If diag = U, all diagonal elements of A are one. A is overwritten with its inverse.

LinearAlgebra.LAPACK.trtrs!Function
trtrs!(uplo, trans, diag, A, B)

Solves A * X = B (trans = N), transpose(A) * X = B (trans = T), or adjoint(A) * X = B (trans = C) for (upper if uplo = U, lower if uplo = L) triangular matrix A. If diag = N, A has non-unit diagonal elements. If diag = U, all diagonal elements of A are one. B is overwritten with the solution X.

LinearAlgebra.LAPACK.trcon!Function
trcon!(norm, uplo, diag, A)

Finds the reciprocal condition number of (upper if uplo = U, lower if uplo = L) triangular matrix A. If diag = N, A has non-unit diagonal elements. If diag = U, all diagonal elements of A are one. If norm = I, the condition number is found in the infinity norm. If norm = O or 1, the condition number is found in the one norm.

LinearAlgebra.LAPACK.trevc!Function
trevc!(side, howmny, select, T, VL = similar(T), VR = similar(T))

Finds the eigensystem of an upper triangular matrix T. If side = R, the right eigenvectors are computed. If side = L, the left eigenvectors are computed. If side = B, both sets are computed. If howmny = A, all eigenvectors are found. If howmny = B, all eigenvectors are found and backtransformed using VL and VR. If howmny = S, only the eigenvectors corresponding to the values in select are computed.

LinearAlgebra.LAPACK.trrfs!Function
trrfs!(uplo, trans, diag, A, B, X, Ferr, Berr) -> (Ferr, Berr)

Estimates the error in the solution to A * X = B (trans = N), transpose(A) * X = B (trans = T), adjoint(A) * X = B (trans = C) for side = L, or the equivalent equations a right-handed side = R X * A after computing X using trtrs!. If uplo = U, A is upper triangular. If uplo = L, A is lower triangular. If diag = N, A has non-unit diagonal elements. If diag = U, all diagonal elements of A are one. Ferr and Berr are optional inputs. Ferr is the forward error and Berr is the backward error, each component-wise.

LinearAlgebra.LAPACK.stev!Function
stev!(job, dv, ev) -> (dv, Zmat)

Computes the eigensystem for a symmetric tridiagonal matrix with dv as diagonal and ev as off-diagonal. If job = N only the eigenvalues are found and returned in dv. If job = V then the eigenvectors are also found and returned in Zmat.

LinearAlgebra.LAPACK.stebz!Function
stebz!(range, order, vl, vu, il, iu, abstol, dv, ev) -> (dv, iblock, isplit)

Computes the eigenvalues for a symmetric tridiagonal matrix with dv as diagonal and ev as off-diagonal. If range = A, all the eigenvalues are found. If range = V, the eigenvalues in the half-open interval (vl, vu] are found. If range = I, the eigenvalues with indices between il and iu are found. If order = B, eigvalues are ordered within a block. If order = E, they are ordered across all the blocks. abstol can be set as a tolerance for convergence.

LinearAlgebra.LAPACK.stegr!Function
stegr!(jobz, range, dv, ev, vl, vu, il, iu) -> (w, Z)

Computes the eigenvalues (jobz = N) or eigenvalues and eigenvectors (jobz = V) for a symmetric tridiagonal matrix with dv as diagonal and ev as off-diagonal. If range = A, all the eigenvalues are found. If range = V, the eigenvalues in the half-open interval (vl, vu] are found. If range = I, the eigenvalues with indices between il and iu are found. The eigenvalues are returned in w and the eigenvectors in Z.

LinearAlgebra.LAPACK.stein!Function
stein!(dv, ev_in, w_in, iblock_in, isplit_in)

Computes the eigenvectors for a symmetric tridiagonal matrix with dv as diagonal and ev_in as off-diagonal. w_in specifies the input eigenvalues for which to find corresponding eigenvectors. iblock_in specifies the submatrices corresponding to the eigenvalues in w_in. isplit_in specifies the splitting points between the submatrix blocks.

LinearAlgebra.LAPACK.syconv!Function
syconv!(uplo, A, ipiv) -> (A, work)

Converts a symmetric matrix A (which has been factorized into a triangular matrix) into two matrices L and D. If uplo = U, A is upper triangular. If uplo = L, it is lower triangular. ipiv is the pivot vector from the triangular factorization. A is overwritten by L and D.

LinearAlgebra.LAPACK.sysv!Function
sysv!(uplo, A, B) -> (B, A, ipiv)

Finds the solution to A * X = B for symmetric matrix A. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. B is overwritten by the solution X. A is overwritten by its Bunch-Kaufman factorization. ipiv contains pivoting information about the factorization.

LinearAlgebra.LAPACK.sytrf!Function
sytrf!(uplo, A) -> (A, ipiv, info)

Computes the Bunch-Kaufman factorization of a symmetric matrix A. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored.

Returns A, overwritten by the factorization, a pivot vector ipiv, and the error code info which is a non-negative integer. If info is positive the matrix is singular and the diagonal part of the factorization is exactly zero at position info.

LinearAlgebra.LAPACK.sytri!Function
sytri!(uplo, A, ipiv)

Computes the inverse of a symmetric matrix A using the results of sytrf!. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. A is overwritten by its inverse.

LinearAlgebra.LAPACK.sytrs!Function
sytrs!(uplo, A, ipiv, B)

Solves the equation A * X = B for a symmetric matrix A using the results of sytrf!. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. B is overwritten by the solution X.

LinearAlgebra.LAPACK.hesv!Function
hesv!(uplo, A, B) -> (B, A, ipiv)

Finds the solution to A * X = B for Hermitian matrix A. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. B is overwritten by the solution X. A is overwritten by its Bunch-Kaufman factorization. ipiv contains pivoting information about the factorization.

LinearAlgebra.LAPACK.hetrf!Function
hetrf!(uplo, A) -> (A, ipiv, info)

Computes the Bunch-Kaufman factorization of a Hermitian matrix A. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored.

Returns A, overwritten by the factorization, a pivot vector ipiv, and the error code info which is a non-negative integer. If info is positive the matrix is singular and the diagonal part of the factorization is exactly zero at position info.

LinearAlgebra.LAPACK.hetri!Function
hetri!(uplo, A, ipiv)

Computes the inverse of a Hermitian matrix A using the results of sytrf!. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. A is overwritten by its inverse.

LinearAlgebra.LAPACK.hetrs!Function
hetrs!(uplo, A, ipiv, B)

Solves the equation A * X = B for a Hermitian matrix A using the results of sytrf!. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. B is overwritten by the solution X.

LinearAlgebra.LAPACK.syev!Function
syev!(jobz, uplo, A)

Finds the eigenvalues (jobz = N) or eigenvalues and eigenvectors (jobz = V) of a symmetric matrix A. If uplo = U, the upper triangle of A is used. If uplo = L, the lower triangle of A is used.

LinearAlgebra.LAPACK.syevr!Function
syevr!(jobz, range, uplo, A, vl, vu, il, iu, abstol) -> (W, Z)

Finds the eigenvalues (jobz = N) or eigenvalues and eigenvectors (jobz = V) of a symmetric matrix A. If uplo = U, the upper triangle of A is used. If uplo = L, the lower triangle of A is used. If range = A, all the eigenvalues are found. If range = V, the eigenvalues in the half-open interval (vl, vu] are found. If range = I, the eigenvalues with indices between il and iu are found. abstol can be set as a tolerance for convergence.

The eigenvalues are returned in W and the eigenvectors in Z.

LinearAlgebra.LAPACK.sygvd!Function
sygvd!(itype, jobz, uplo, A, B) -> (w, A, B)

Finds the generalized eigenvalues (jobz = N) or eigenvalues and eigenvectors (jobz = V) of a symmetric matrix A and symmetric positive-definite matrix B. If uplo = U, the upper triangles of A and B are used. If uplo = L, the lower triangles of A and B are used. If itype = 1, the problem to solve is A * x = lambda * B * x. If itype = 2, the problem to solve is A * B * x = lambda * x. If itype = 3, the problem to solve is B * A * x = lambda * x.

LinearAlgebra.LAPACK.bdsqr!Function
bdsqr!(uplo, d, e_, Vt, U, C) -> (d, Vt, U, C)

Computes the singular value decomposition of a bidiagonal matrix with d on the diagonal and e_ on the off-diagonal. If uplo = U, e_ is the superdiagonal. If uplo = L, e_ is the subdiagonal. Can optionally also compute the product Q' * C.

Returns the singular values in d, and the matrix C overwritten with Q' * C.

LinearAlgebra.LAPACK.bdsdc!Function
bdsdc!(uplo, compq, d, e_) -> (d, e, u, vt, q, iq)

Computes the singular value decomposition of a bidiagonal matrix with d on the diagonal and e_ on the off-diagonal using a divide and conqueq method. If uplo = U, e_ is the superdiagonal. If uplo = L, e_ is the subdiagonal. If compq = N, only the singular values are found. If compq = I, the singular values and vectors are found. If compq = P, the singular values and vectors are found in compact form. Only works for real types.

Returns the singular values in d, and if compq = P, the compact singular vectors in iq.

LinearAlgebra.LAPACK.gecon!Function
gecon!(normtype, A, anorm)

Finds the reciprocal condition number of matrix A. If normtype = I, the condition number is found in the infinity norm. If normtype = O or 1, the condition number is found in the one norm. A must be the result of getrf! and anorm is the norm of A in the relevant norm.

LinearAlgebra.LAPACK.gehrd!Function
gehrd!(ilo, ihi, A) -> (A, tau)

Converts a matrix A to Hessenberg form. If A is balanced with gebal! then ilo and ihi are the outputs of gebal!. Otherwise they should be ilo = 1 and ihi = size(A,2). tau contains the elementary reflectors of the factorization.

LinearAlgebra.LAPACK.orghr!Function
orghr!(ilo, ihi, A, tau)

Explicitly finds Q, the orthogonal/unitary matrix from gehrd!. ilo, ihi, A, and tau must correspond to the input/output to gehrd!.

LinearAlgebra.LAPACK.gees!Function
gees!(jobvs, A) -> (A, vs, w)

Computes the eigenvalues (jobvs = N) or the eigenvalues and Schur vectors (jobvs = V) of matrix A. A is overwritten by its Schur form.

Returns A, vs containing the Schur vectors, and w, containing the eigenvalues.

LinearAlgebra.LAPACK.gges!Function
gges!(jobvsl, jobvsr, A, B) -> (A, B, alpha, beta, vsl, vsr)

Computes the generalized eigenvalues, generalized Schur form, left Schur vectors (jobsvl = V), or right Schur vectors (jobvsr = V) of A and B.

The generalized eigenvalues are returned in alpha and beta. The left Schur vectors are returned in vsl and the right Schur vectors are returned in vsr.

LinearAlgebra.LAPACK.trexc!Function
trexc!(compq, ifst, ilst, T, Q) -> (T, Q)

Reorder the Schur factorization of a matrix. If compq = V, the Schur vectors Q are reordered. If compq = N they are not modified. ifst and ilst specify the reordering of the vectors.

LinearAlgebra.LAPACK.trsen!Function
trsen!(compq, job, select, T, Q) -> (T, Q, w, s, sep)

Reorder the Schur factorization of a matrix and optionally finds reciprocal condition numbers. If job = N, no condition numbers are found. If job = E, only the condition number for this cluster of eigenvalues is found. If job = V, only the condition number for the invariant subspace is found. If job = B then the condition numbers for the cluster and subspace are found. If compq = V the Schur vectors Q are updated. If compq = N the Schur vectors are not modified. select determines which eigenvalues are in the cluster.

Returns T, Q, reordered eigenvalues in w, the condition number of the cluster of eigenvalues s, and the condition number of the invariant subspace sep.

LinearAlgebra.LAPACK.tgsen!Function
tgsen!(select, S, T, Q, Z) -> (S, T, alpha, beta, Q, Z)

Reorders the vectors of a generalized Schur decomposition. select specifies the eigenvalues in each cluster.

LinearAlgebra.LAPACK.trsyl!Function
trsyl!(transa, transb, A, B, C, isgn=1) -> (C, scale)

Solves the Sylvester matrix equation A * X +/- X * B = scale*C where A and B are both quasi-upper triangular. If transa = N, A is not modified. If transa = T, A is transposed. If transa = C, A is conjugate transposed. Similarly for transb and B. If isgn = 1, the equation A * X + X * B = scale * C is solved. If isgn = -1, the equation A * X - X * B = scale * C is solved.

Returns X (overwriting C) and scale.

• Bischof1987C Bischof and C Van Loan, "The WY representation for products of Householder matrices", SIAM J Sci Stat Comput 8 (1987), s2-s13. doi:10.1137/0908009
• Schreiber1989R Schreiber and C Van Loan, "A storage-efficient WY representation for products of Householder transformations", SIAM J Sci Stat Comput 10 (1989), 53-57. doi:10.1137/0910005
• Bunch1977J R Bunch and L Kaufman, Some stable methods for calculating inertia and solving symmetric linear systems, Mathematics of Computation 31:137 (1977), 163-179. url.
• issue8859Issue 8859, "Fix least squares", https://github.com/JuliaLang/julia/pull/8859
• B96Åke Björck, "Numerical Methods for Least Squares Problems", SIAM Press, Philadelphia, 1996, "Other Titles in Applied Mathematics", Vol. 51. doi:10.1137/1.9781611971484
• S84G. W. Stewart, "Rank Degeneracy", SIAM Journal on Scientific and Statistical Computing, 5(2), 1984, 403-413. doi:10.1137/0905030
• KY88Konstantinos Konstantinides and Kung Yao, "Statistical analysis of effective singular values in matrix rank determination", IEEE Transactions on Acoustics, Speech and Signal Processing, 36(5), 1988, 757-763. doi:10.1109/29.1585
• H05Nicholas J. Higham, "The squaring and scaling method for the matrix exponential revisited", SIAM Journal on Matrix Analysis and Applications, 26(4), 2005, 1179-1193. doi:10.1137/090768539
• AH12Awad H. Al-Mohy and Nicholas J. Higham, "Improved inverse scaling and squaring algorithms for the matrix logarithm", SIAM Journal on Scientific Computing, 34(4), 2012, C153-C169. doi:10.1137/110852553
• AHR13Awad H. Al-Mohy, Nicholas J. Higham and Samuel D. Relton, "Computing the Fréchet derivative of the matrix logarithm and estimating the condition number", SIAM Journal on Scientific Computing, 35(4), 2013, C394-C410. doi:10.1137/120885991
• BH83Åke Björck and Sven Hammarling, "A Schur method for the square root of a matrix", Linear Algebra and its Applications, 52-53, 1983, 127-140. doi:10.1016/0024-3795(83)80010-X
• AH16_1Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577
• AH16_2Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577
• AH16_3Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577
• AH16_4Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577
• AH16_5Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577
• AH16_6Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577