Julia 线性代数

线性代数

DocTestSetup = :(using LinearAlgebra)

除了对多维数组的支持（并作为其一部分），Julia还提供了例子。

许多常见和有用的线性代数运算，可以使用LinearAlgebra进行加载。 基本操作，例如 tr, det, inv都支持：

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

以及其它有用的操作，例如寻找特征值或特征向量：

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

此外，Julia提供了许多 [factorizations](@ref man-linalg-factorizations)，可用于通过将矩阵分解的形式来加快线性求解或矩阵求幂等问题（出于性能或内存的考虑）更适合该问题。 请参阅有关 factorize的文档想要了解更多的信息。

举个例子：

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

由于A不是Hermitian，对称，三角形，三对角或bidiagonal，因此LU分解可能是我们能做的最好的。

与之比较：

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

在这里，Julia能够检测到B实际上是对称的，并使用了更合适的因式分解。通常可以为已知具有某些属性的矩阵编写更有效的代码，例如它是对称矩阵或三对角矩阵。 Julia提供了一些特殊类型，以便您可以将矩阵“tag”为具有这些属性。

例如：

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 已被标记为（真实）对称的矩阵，因此对于以后的操作，我们可能会对其执行，例如本征分解或计算矩阵向量乘积，效率提高一半。

例如：

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

这里的\操作执行线性求解。 左除运算符功能强大，并且易于编写紧凑，易读的代码，该代码足够灵活以解决各种线性方程组。

特殊矩阵

Matrices with special symmetries and structures在线性代数中经常出现，并且经常与各种矩阵分解有关。

Julia具有大量特殊矩阵类型的集合，这些属性允许使用专门针对特定矩阵类型开发的专用例程进行快速计算。

下表总结了在Julia中实现的特殊矩阵的类型，以及在LAPACK中是否可以使用针对各种优化方法的挂钩。

类型	描述
Symmetric	Symmetric matrix
Hermitian	Hermitian matrix
UpperTriangular	Upper triangular matrix
UnitUpperTriangular	Upper triangular matrix with unit diagonal
LowerTriangular	Lower triangular matrix
UnitLowerTriangular	Lower triangular matrix with unit diagonal

| UpperHessenberg | Upper Hessenberg matrix
| Tridiagonal | Tridiagonal matrix |
| SymTridiagonal | Symmetric tridiagonal matrix |
| Bidiagonal | Upper/lower bidiagonal matrix |
| Diagonal | Diagonal matrix |
| UniformScaling | Uniform scaling operator |

基本操作

矩阵类型	+	-	*	\	其它函数及优化方法
Symmetric				MV	inv, sqrt, exp
Hermitian				MV	inv, sqrt, exp
UpperTriangular			MV	MV	inv, det
UnitUpperTriangular			MV	MV	inv, det
LowerTriangular			MV	MV	inv, det
UnitLowerTriangular			MV	MV	inv, det
UpperHessenberg				MM	inv, det
SymTridiagonal	M	M	MS	MV	eigmax, eigmin
Tridiagonal	M	M	MS	MV
Bidiagonal	M	M	MS	MV
Diagonal	M	M	MV	MV	inv, det, logdet, /
UniformScaling	M	M	MVS	MVS	/

Legend:

Key	描述
M (矩阵)	提供了矩阵-矩阵运算的优化方法
V (向量)	提供了矩阵-向量运算的优化方法
S (标量)	提供了矩阵-标量运算的优化方法

矩阵分解

矩阵类型	LAPACK	eigen	eigvals	eigvecs	svd	svdvals
Symmetric	SY		ARI
Hermitian	HE		ARI
UpperTriangular	TR	A	A	A
UnitUpperTriangular	TR	A	A	A
LowerTriangular	TR	A	A	A
UnitLowerTriangular	TR	A	A	A
SymTridiagonal	ST	A	ARI	AV
Tridiagonal	GT
Bidiagonal	BD				A	A
Diagonal	DI		A

Legend:

Key	描述	例子
A (all)	提供了寻找所有特征值和/或向量的优化方法	e.g. eigvals(M)
R (range)	提供了通过ih特征值找到il的优化方法	eigvals(M, il, ih)
I (interval)	提供了一种在区间[vl，vh]中查找特征值的优化方法	eigvals(M, vl, vh)
V (vectors)	提供了一种找到与特征值x = [x1，x2，…]对应的特征向量的优化方法	eigvecs(M, x)

UniformScaling运算符

UniformScaling 运算符表示标量乘以标识运算符λ* I。 单位运算符I定义为常量，并且是UniformScaling的实例。 这些运算符的大小是通用的，并且与二进制运算符+，-，*和\中的其他矩阵匹配。 对于A + I和A-I，这意味着A必须为正方形。 与单位运算符I的乘积是无操作的（检查比例因子是否为1除外），因此几乎没有开销。要查看运行中的UniformScaling运算符：

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:
[...]

如果您需要针对相同的A和不同的μ求解许多形式为（A +μI）x = b的系统，那么有助于通过 hessenberg 函数计算A的Hessenberg分解F。给定F，Julia对（F +μ* I）\ b（相当于（A +μ* I）x \ b）和相关操作（如行列式）采用了有效的算法。

[矩阵分解](@id man-linalg-factorizations)

矩阵分解将矩阵分解为矩阵的乘积，并且是线性代数的核心概念之一。下表总结了已在Julia中实现的矩阵分解的类型。 有关其相关方法的详细信息，请参见线性代数文档的Standard functions 部分。

Type	Description
BunchKaufman	Bunch-Kaufman factorization
Cholesky	Cholesky factorization
CholeskyPivoted	Pivoted Cholesky factorization
LDLt	LDL(T) factorization
LU	LU factorization
QR	QR factorization
QRCompactWY	Compact WY form of the QR factorization
QRPivoted	Pivoted QR factorization
LQ	QR factorization of transpose(A)
Hessenberg	Hessenberg decomposition
Eigen	Spectral decomposition
GeneralizedEigen	Generalized spectral decomposition
SVD	Singular value decomposition
GeneralizedSVD	Generalized SVD
Schur	Schur decomposition
GeneralizedSchur	Generalized Schur decomposition

标准函数

Julia中的线性代数函数主要通过调用LAPACK来实现。

SuiteSparse中的稀疏分解调用函数。

Base.:*(::AbstractMatrix, ::AbstractMatrix)
Base.:\(::AbstractMatrix, ::AbstractVecOrMat)
LinearAlgebra.SingularException
LinearAlgebra.PosDefException
LinearAlgebra.ZeroPivotException
LinearAlgebra.dot
LinearAlgebra.cross
LinearAlgebra.factorize
LinearAlgebra.Diagonal
LinearAlgebra.Bidiagonal
LinearAlgebra.SymTridiagonal
LinearAlgebra.Tridiagonal
LinearAlgebra.Symmetric
LinearAlgebra.Hermitian
LinearAlgebra.LowerTriangular
LinearAlgebra.UpperTriangular
LinearAlgebra.UnitLowerTriangular
LinearAlgebra.UnitUpperTriangular
LinearAlgebra.UpperHessenberg
LinearAlgebra.UniformScaling
LinearAlgebra.I
LinearAlgebra.Factorization
LinearAlgebra.LU
LinearAlgebra.lu
LinearAlgebra.lu!
LinearAlgebra.Cholesky
LinearAlgebra.CholeskyPivoted
LinearAlgebra.cholesky
LinearAlgebra.cholesky!
LinearAlgebra.lowrankupdate
LinearAlgebra.lowrankdowndate
LinearAlgebra.lowrankupdate!
LinearAlgebra.lowrankdowndate!
LinearAlgebra.LDLt
LinearAlgebra.ldlt
LinearAlgebra.ldlt!
LinearAlgebra.QR
LinearAlgebra.QRCompactWY
LinearAlgebra.QRPivoted
LinearAlgebra.qr
LinearAlgebra.qr!
LinearAlgebra.LQ
LinearAlgebra.lq
LinearAlgebra.lq!
LinearAlgebra.BunchKaufman
LinearAlgebra.bunchkaufman
LinearAlgebra.bunchkaufman!
LinearAlgebra.Eigen
LinearAlgebra.GeneralizedEigen
LinearAlgebra.eigvals
LinearAlgebra.eigvals!
LinearAlgebra.eigmax
LinearAlgebra.eigmin
LinearAlgebra.eigvecs
LinearAlgebra.eigen
LinearAlgebra.eigen!
LinearAlgebra.Hessenberg
LinearAlgebra.hessenberg
LinearAlgebra.hessenberg!
LinearAlgebra.Schur
LinearAlgebra.GeneralizedSchur
LinearAlgebra.schur
LinearAlgebra.schur!
LinearAlgebra.ordschur
LinearAlgebra.ordschur!
LinearAlgebra.SVD
LinearAlgebra.GeneralizedSVD
LinearAlgebra.svd
LinearAlgebra.svd!
LinearAlgebra.svdvals
LinearAlgebra.svdvals!
LinearAlgebra.Givens
LinearAlgebra.givens
LinearAlgebra.triu
LinearAlgebra.triu!
LinearAlgebra.tril
LinearAlgebra.tril!
LinearAlgebra.diagind
LinearAlgebra.diag
LinearAlgebra.diagm
LinearAlgebra.rank
LinearAlgebra.norm
LinearAlgebra.opnorm
LinearAlgebra.normalize!
LinearAlgebra.normalize
LinearAlgebra.cond
LinearAlgebra.condskeel
LinearAlgebra.tr
LinearAlgebra.det
LinearAlgebra.logdet
LinearAlgebra.logabsdet
Base.inv(::AbstractMatrix)
LinearAlgebra.pinv
LinearAlgebra.nullspace
Base.kron
LinearAlgebra.exp(::StridedMatrix{<:LinearAlgebra.BlasFloat})
Base.:^(::AbstractMatrix, ::Number)
Base.:^(::Number, ::AbstractMatrix)
LinearAlgebra.log(::StridedMatrix)
LinearAlgebra.sqrt(::StridedMatrix{<:Real})
LinearAlgebra.cos(::StridedMatrix{<:Real})
LinearAlgebra.sin(::StridedMatrix{<:Real})
LinearAlgebra.sincos(::StridedMatrix{<:Real})
LinearAlgebra.tan(::StridedMatrix{<:Real})
LinearAlgebra.sec(::StridedMatrix)
LinearAlgebra.csc(::StridedMatrix)
LinearAlgebra.cot(::StridedMatrix)
LinearAlgebra.cosh(::StridedMatrix)
LinearAlgebra.sinh(::StridedMatrix)
LinearAlgebra.tanh(::StridedMatrix)
LinearAlgebra.sech(::StridedMatrix)
LinearAlgebra.csch(::StridedMatrix)
LinearAlgebra.coth(::StridedMatrix)
LinearAlgebra.acos(::StridedMatrix)
LinearAlgebra.asin(::StridedMatrix)
LinearAlgebra.atan(::StridedMatrix)
LinearAlgebra.asec(::StridedMatrix)
LinearAlgebra.acsc(::StridedMatrix)
LinearAlgebra.acot(::StridedMatrix)
LinearAlgebra.acosh(::StridedMatrix)
LinearAlgebra.asinh(::StridedMatrix)
LinearAlgebra.atanh(::StridedMatrix)
LinearAlgebra.asech(::StridedMatrix)
LinearAlgebra.acsch(::StridedMatrix)
LinearAlgebra.acoth(::StridedMatrix)
LinearAlgebra.lyap
LinearAlgebra.sylvester
LinearAlgebra.issuccess
LinearAlgebra.issymmetric
LinearAlgebra.isposdef
LinearAlgebra.isposdef!
LinearAlgebra.istril
LinearAlgebra.istriu
LinearAlgebra.isdiag
LinearAlgebra.ishermitian
Base.transpose
LinearAlgebra.transpose!
LinearAlgebra.Transpose
Base.adjoint
LinearAlgebra.adjoint!
LinearAlgebra.Adjoint
Base.copy(::Union{Transpose,Adjoint})
LinearAlgebra.stride1
LinearAlgebra.checksquare
LinearAlgebra.peakflops

低阶矩阵运算

在许多情况下，存在矩阵操作的内置版本，可让您提供预分配的输出矢量或矩阵。 在优化关键代码以避免重复分配的开销时，这很有用。 根据通常的Julia约定，这些就地操作的后缀为！（例如mul！）。

LinearAlgebra.mul!
LinearAlgebra.lmul!
LinearAlgebra.rmul!
LinearAlgebra.ldiv!
LinearAlgebra.rdiv!

BLAS 函数

在Julia中（与许多科学计算一样），密集的线性代数运算基于LAPACK库（高性能的线性代数计算库），而LAPACK库又建立在称为BLAS的基本线性代数构件之上。

每种计算机体系结构都有高度优化的BLAS实现，有时在高性能线性代数例程中，直接调用BLAS函数很有用。

LinearAlgebra.BLAS为某些BLAS函数提供包装器。

• 那些重写输入数组的BLAS函数的名称以'！'结尾。
• 通常，BLAS函数定义了4种方法，每种方法分别用于Float64，Float32，ComplexF64和ComplexF32数组。

[BLAS 参数](@id stdlib-blas-chars)

许多BLAS函数都接受参数，这些参数确定是否要转置（trans），要引用的矩阵的哪个三角形（uplo或ul），是否可以将三角形矩阵的对角线假定为全1（dA）或该矩阵乘法的顺序（side）。

可能是：

[乘法顺序](@id stdlib-blas-side)

side	Meaning
'L'	该参数位于矩阵-矩阵运算的左侧。
'R'	该参数位于矩阵-矩阵运算的右侧。

[三角矩阵参考](@id stdlib-blas-uplo)

uplo/ul	Meaning
'U'	只能在上三角矩阵中使用
'L'	只能在下三角矩阵中使用

[转置](@id stdlib-blas-trans)

trans/tX	Meaning
'N'	输入矩阵X不转置或共轭。
'T'	输入矩阵X将转置。
'C'	输入矩阵X将被共轭和转置。

[Unit diagonal](@id stdlib-blas-diag)

diag/dX	Meaning
'N'	读取矩阵X的对角线值。
'U'	假设矩阵X的对角线全为1。

LinearAlgebra.BLAS
LinearAlgebra.BLAS.dot
LinearAlgebra.BLAS.dotu
LinearAlgebra.BLAS.dotc
LinearAlgebra.BLAS.blascopy!
LinearAlgebra.BLAS.nrm2
LinearAlgebra.BLAS.asum
LinearAlgebra.axpy!
LinearAlgebra.axpby!
LinearAlgebra.BLAS.scal!
LinearAlgebra.BLAS.scal
LinearAlgebra.BLAS.iamax
LinearAlgebra.BLAS.ger!
LinearAlgebra.BLAS.syr!
LinearAlgebra.BLAS.syrk!
LinearAlgebra.BLAS.syrk
LinearAlgebra.BLAS.syr2k!
LinearAlgebra.BLAS.syr2k
LinearAlgebra.BLAS.her!
LinearAlgebra.BLAS.herk!
LinearAlgebra.BLAS.herk
LinearAlgebra.BLAS.her2k!
LinearAlgebra.BLAS.her2k
LinearAlgebra.BLAS.gbmv!
LinearAlgebra.BLAS.gbmv
LinearAlgebra.BLAS.sbmv!
LinearAlgebra.BLAS.sbmv(::Any, ::Any, ::Any, ::Any, ::Any)
LinearAlgebra.BLAS.sbmv(::Any, ::Any, ::Any, ::Any)
LinearAlgebra.BLAS.gemm!
LinearAlgebra.BLAS.gemm(::Any, ::Any, ::Any, ::Any, ::Any)
LinearAlgebra.BLAS.gemm(::Any, ::Any, ::Any, ::Any)
LinearAlgebra.BLAS.gemv!
LinearAlgebra.BLAS.gemv(::Any, ::Any, ::Any, ::Any)
LinearAlgebra.BLAS.gemv(::Any, ::Any, ::Any)
LinearAlgebra.BLAS.symm!
LinearAlgebra.BLAS.symm(::Any, ::Any, ::Any, ::Any, ::Any)
LinearAlgebra.BLAS.symm(::Any, ::Any, ::Any, ::Any)
LinearAlgebra.BLAS.symv!
LinearAlgebra.BLAS.symv(::Any, ::Any, ::Any, ::Any)
LinearAlgebra.BLAS.symv(::Any, ::Any, ::Any)
LinearAlgebra.BLAS.hemm!
LinearAlgebra.BLAS.hemm(::Any, ::Any, ::Any, ::Any, ::Any)
LinearAlgebra.BLAS.hemm(::Any, ::Any, ::Any, ::Any)
LinearAlgebra.BLAS.hemv!
LinearAlgebra.BLAS.hemv(::Any, ::Any, ::Any, ::Any)
LinearAlgebra.BLAS.hemv(::Any, ::Any, ::Any)
LinearAlgebra.BLAS.trmm!
LinearAlgebra.BLAS.trmm
LinearAlgebra.BLAS.trsm!
LinearAlgebra.BLAS.trsm
LinearAlgebra.BLAS.trmv!
LinearAlgebra.BLAS.trmv
LinearAlgebra.BLAS.trsv!
LinearAlgebra.BLAS.trsv
LinearAlgebra.BLAS.set_num_threads

LAPACK 函数

LinearAlgebra.LAPACK 可用于计算诸如求解线性代数方程、线性系统方程组的最小平方解、计算特征值和特征向量等问题。

• 重写的输入函数名称都已'！'结尾。
• 通常，一个函数定义了4种方法，每种方法分别用于Float64，Float32，ComplexF64和ComplexF32数组。

请注意，Julia提供的LAPACK API将来可能会更改。 由于此API不是面向用户的，因此在未来的发行版中不承诺支持/弃用此特定功能集。

LinearAlgebra.LAPACK
LinearAlgebra.LAPACK.gbtrf!
LinearAlgebra.LAPACK.gbtrs!
LinearAlgebra.LAPACK.gebal!
LinearAlgebra.LAPACK.gebak!
LinearAlgebra.LAPACK.gebrd!
LinearAlgebra.LAPACK.gelqf!
LinearAlgebra.LAPACK.geqlf!
LinearAlgebra.LAPACK.geqrf!
LinearAlgebra.LAPACK.geqp3!
LinearAlgebra.LAPACK.gerqf!
LinearAlgebra.LAPACK.geqrt!
LinearAlgebra.LAPACK.geqrt3!
LinearAlgebra.LAPACK.getrf!
LinearAlgebra.LAPACK.tzrzf!
LinearAlgebra.LAPACK.ormrz!
LinearAlgebra.LAPACK.gels!
LinearAlgebra.LAPACK.gesv!
LinearAlgebra.LAPACK.getrs!
LinearAlgebra.LAPACK.getri!
LinearAlgebra.LAPACK.gesvx!
LinearAlgebra.LAPACK.gelsd!
LinearAlgebra.LAPACK.gelsy!
LinearAlgebra.LAPACK.gglse!
LinearAlgebra.LAPACK.geev!
LinearAlgebra.LAPACK.gesdd!
LinearAlgebra.LAPACK.gesvd!
LinearAlgebra.LAPACK.ggsvd!
LinearAlgebra.LAPACK.ggsvd3!
LinearAlgebra.LAPACK.geevx!
LinearAlgebra.LAPACK.ggev!
LinearAlgebra.LAPACK.gtsv!
LinearAlgebra.LAPACK.gttrf!
LinearAlgebra.LAPACK.gttrs!
LinearAlgebra.LAPACK.orglq!
LinearAlgebra.LAPACK.orgqr!
LinearAlgebra.LAPACK.orgql!
LinearAlgebra.LAPACK.orgrq!
LinearAlgebra.LAPACK.ormlq!
LinearAlgebra.LAPACK.ormqr!
LinearAlgebra.LAPACK.ormql!
LinearAlgebra.LAPACK.ormrq!
LinearAlgebra.LAPACK.gemqrt!
LinearAlgebra.LAPACK.posv!
LinearAlgebra.LAPACK.potrf!
LinearAlgebra.LAPACK.potri!
LinearAlgebra.LAPACK.potrs!
LinearAlgebra.LAPACK.pstrf!
LinearAlgebra.LAPACK.ptsv!
LinearAlgebra.LAPACK.pttrf!
LinearAlgebra.LAPACK.pttrs!
LinearAlgebra.LAPACK.trtri!
LinearAlgebra.LAPACK.trtrs!
LinearAlgebra.LAPACK.trcon!
LinearAlgebra.LAPACK.trevc!
LinearAlgebra.LAPACK.trrfs!
LinearAlgebra.LAPACK.stev!
LinearAlgebra.LAPACK.stebz!
LinearAlgebra.LAPACK.stegr!
LinearAlgebra.LAPACK.stein!
LinearAlgebra.LAPACK.syconv!
LinearAlgebra.LAPACK.sysv!
LinearAlgebra.LAPACK.sytrf!
LinearAlgebra.LAPACK.sytri!
LinearAlgebra.LAPACK.sytrs!
LinearAlgebra.LAPACK.hesv!
LinearAlgebra.LAPACK.hetrf!
LinearAlgebra.LAPACK.hetri!
LinearAlgebra.LAPACK.hetrs!
LinearAlgebra.LAPACK.syev!
LinearAlgebra.LAPACK.syevr!
LinearAlgebra.LAPACK.sygvd!
LinearAlgebra.LAPACK.bdsqr!
LinearAlgebra.LAPACK.bdsdc!
LinearAlgebra.LAPACK.gecon!
LinearAlgebra.LAPACK.gehrd!
LinearAlgebra.LAPACK.orghr!
LinearAlgebra.LAPACK.gees!
LinearAlgebra.LAPACK.gges!
LinearAlgebra.LAPACK.trexc!
LinearAlgebra.LAPACK.trsen!
LinearAlgebra.LAPACK.tgsen!
LinearAlgebra.LAPACK.trsyl!

DocTestSetup = nothing