Workspaces

Workspaces represent the buffers and temporary storage that are used during the computations of LAPACK functions. Upon initialization with a template matrix, work buffers will be allocated that are appropriate to be used during the factorization of matrices similar to the template, e.g. both Float64 and Float32 Matrices work, but also Complex numbers are allowed when appropriate.

Workspace

The following convenience function is supplied in order to construct the correct Workspace for a given LAPACK function. This can then be used to perform the decompositions without extra allocations.

FastLapackInterface.Workspace — Type

Workspace(lapack_function, A)

Will create the correct Workspace for the target lapack_function and matrix A.

Examples

julia> A = [1.2 2.3
            6.2 3.3]
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

julia> ws = Workspace(LAPACK.geqrt!, A)
QRWYWs{Float64, Matrix{Float64}}
  work: 4-element Vector{Float64}
  T: 2×2 Matrix{Float64}


julia> LinearAlgebra.QRCompactWY(factorize!(ws, A)...)
LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}
Q factor: 2×2 LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}
R factor:
2×2 Matrix{Float64}:
 -6.31506  -3.67692
  0.0      -1.63102

Each Workspace also has a function to resize! to allow for its use with larger matrices or with more features (e.g. the computation of left eigenvectors and right eigenvectors using EigenWs).

Base.resize! — Method

resize!(ws, A; kwargs...)

Resizes the ws to be appropriate for use with matrix A. The kwargs can be used to communicate which features should be supported by the Workspace, such as left and right eigenvectors while using EigenWs. This function is mainly used for automatic resizing inside LAPACK functions.

BunchKaufman

FastLapackInterface.BunchKaufmanWs — Type

BunchKaufmanWs

Workspace for LinearAlgebra.BunchKaufman factorization using the LAPACK.sytrf! or LAPACK.sytrf_rook! functions for symmetric matrices, and LAPACK.hetrf! or LAPACK.hetrf_rook! functions for hermitian matrices (e.g. with ComplexF64 or ComplexF32 elements).

Examples

julia> A = [1.2 7.8
            7.8 3.3]
2×2 Matrix{Float64}:
 1.2  7.8
 7.8  3.3

julia> ws = BunchKaufmanWs(A)
BunchKaufmanWs{Float64}
  work: 128-element Vector{Float64}
  ipiv: 2-element Vector{Int64}


julia> A, ipiv, info = LAPACK.sytrf!(ws, 'U', A)
([1.2 7.8; 7.8 3.3], [-1, -1], 0)

julia> t = LinearAlgebra.BunchKaufman(A, ipiv,'U', true, false, info)
BunchKaufman{Float64, Matrix{Float64}, Vector{Int64}}
D factor:
2×2 Tridiagonal{Float64, Vector{Float64}}:
 1.2  7.8
 7.8  3.3
U factor:
2×2 UnitUpperTriangular{Float64, Matrix{Float64}}:
 1.0  0.0
  ⋅   1.0
permutation:
2-element Vector{Int64}:
 1
 2

julia> A = [1.2 7.8
            7.8 3.3]
2×2 Matrix{Float64}:
 1.2  7.8
 7.8  3.3

julia> ws = BunchKaufmanWs(A)
BunchKaufmanWs{Float64}
  work: 128-element Vector{Float64}
  ipiv: 2-element Vector{Int64}


julia> A, ipiv, info = LAPACK.sytrf_rook!(ws, 'U', A)
([1.2 7.8; 7.8 3.3], [-1, -2], 0)

julia> t = LinearAlgebra.BunchKaufman(A, ipiv,'U', true, true, info)
BunchKaufman{Float64, Matrix{Float64}, Vector{Int64}}
D factor:
2×2 Tridiagonal{Float64, Vector{Float64}}:
 1.2  7.8
 7.8  3.3
U factor:
2×2 UnitUpperTriangular{Float64, Matrix{Float64}}:
 1.0  0.0
  ⋅   1.0
permutation:
2-element Vector{Int64}:
 1
 2

Cholesky

FastLapackInterface.CholeskyPivotedWs — Type

CholeskyPivotedWs

Workspace for LinearAlgebra.CholeskyPivoted factorization using the LAPACK.pstrf! function. The standard LinearAlgebra.Cholesky uses LAPACK.potrf! which is non-allocating and does not require a separate Workspace.

Examples

julia> A = [1.2 7.8
            7.8 3.3]
2×2 Matrix{Float64}:
 1.2  7.8
 7.8  3.3

julia> ws = CholeskyPivotedWs(A)
CholeskyPivotedWs{Float64}
  work: 4-element Vector{Float64}
  piv: 2-element Vector{Int64}


julia> AA, piv, rank, info = LAPACK.pstrf!(ws, 'U', A, 1e-6)
([1.816590212458495 4.293758683992806; 7.8 -17.236363636363635], [2, 1], 1, 1)

julia> CholeskyPivoted(AA, 'U', piv, rank, 1e-6, info)
CholeskyPivoted{Float64, Matrix{Float64}, Vector{Int64}}
U factor with rank 1:
2×2 UpperTriangular{Float64, Matrix{Float64}}:
 1.81659    4.29376
  ⋅       -17.2364
permutation:
2-element Vector{Int64}:
 2
 1

Eigen

FastLapackInterface.EigenWs — Type

EigenWs

Workspace for LinearAlgebra.Eigen factorization using the LAPACK.geevx! function.

Examples

julia> A = [1.2 2.3
            6.2 3.3]
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

julia> ws = EigenWs(A, rvecs=true)
EigenWs{Float64, Matrix{Float64}, Float64}
  work: 260-element Vector{Float64}
  rwork: 2-element Vector{Float64}
  VL: 0×2 Matrix{Float64}
  VR: 2×2 Matrix{Float64}
  W: 2-element Vector{Float64}
  scale: 2-element Vector{Float64}
  iwork: 0-element Vector{Int64}
  rconde: 0-element Vector{Float64}
  rcondv: 0-element Vector{Float64}


julia> t = LAPACK.geevx!(ws, 'N', 'N', 'V', 'N', A);

julia> LinearAlgebra.Eigen(t[2], t[5])
Eigen{Float64, Float64, Matrix{Float64}, Vector{Float64}}
values:
2-element Vector{Float64}:
 -1.6695025194532018
  6.169502519453203
vectors:
2×2 Matrix{Float64}:
 -0.625424  -0.420019
  0.780285  -0.907515

FastLapackInterface.HermitianEigenWs — Type

HermitianEigenWs

Workspace to be used with Hermitian diagonalization using the LAPACK.syevr! function. Supports both Real and Complex Hermitian matrices.

Examples

julia> A = [1.2 2.3
            6.2 3.3]
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

julia> ws = HermitianEigenWs(A, vecs=true)
HermitianEigenWs{Float64, Matrix{Float64}, Float64}
  work: 66-element Vector{Float64}
  rwork: 0-element Vector{Float64}
  iwork: 20-element Vector{Int64}
  w: 2-element Vector{Float64}
  Z: 2×2 Matrix{Float64}
  isuppz: 4-element Vector{Int64}


julia> LinearAlgebra.Eigen(LAPACK.syevr!(ws, 'V', 'A', 'U', A, 0.0, 0.0, 0, 0, 1e-6)...)
Eigen{Float64, Float64, Matrix{Float64}, Vector{Float64}}
values:
2-element Vector{Float64}:
 -0.2783393759541063
  4.778339375954106
vectors:
2×2 Matrix{Float64}:
 -0.841217  0.540698
  0.540698  0.841217

FastLapackInterface.GeneralizedEigenWs — Type

GeneralizedEigenWs

Workspace that can be used for LinearAlgebra.GeneralizedEigen factorization using LAPACK.ggev!. Supports Real and Complex matrices.

Examples

julia> A = [1.2 2.3
            6.2 3.3]
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

julia> B = [8.2 1.7
            5.9 2.1]
2×2 Matrix{Float64}:
 8.2  1.7
 5.9  2.1

julia> ws = GeneralizedEigenWs(A, rvecs=true)
GeneralizedEigenWs{Float64, Matrix{Float64}, Float64}
  work: 78-element Vector{Float64}
  vl: 0×2 Matrix{Float64}
  vr: 2×2 Matrix{Float64}
  αr: 2-element Vector{Float64}
  αi: 2-element Vector{Float64}
  β: 2-element Vector{Float64}


julia> αr, αi, β, _, vr = LAPACK.ggev!(ws, 'N', 'V', A, B);

julia> LinearAlgebra.GeneralizedEigen(αr ./ β, vr)
GeneralizedEigen{Float64, Float64, Matrix{Float64}, Vector{Float64}}
values:
2-element Vector{Float64}:
 -0.8754932558185097
  1.6362721153456299
vectors:
2×2 Matrix{Float64}:
 -0.452121  -0.0394242
  1.0        1.0

LSE

FastLapackInterface.LSEWs — Type

LSEWs

Workspace for the least squares solving function LAPACK.geqrf!.

Examples

julia> A = [1.2 2.3 6.2
            6.2 3.3 8.8
            9.1 2.1 5.5]
3×3 Matrix{Float64}:
 1.2  2.3  6.2
 6.2  3.3  8.8
 9.1  2.1  5.5

julia> B = [2.7 3.1 7.7
            4.1 8.1 1.8]
2×3 Matrix{Float64}:
 2.7  3.1  7.7
 4.1  8.1  1.8

julia> c = [0.2, 7.2, 2.9]
3-element Vector{Float64}:
 0.2
 7.2
 2.9

julia> d = [3.9, 2.1]
2-element Vector{Float64}:
 3.9
 2.1

julia> ws = LSEWs(A, B)
LSEWs{Float64}
  work: 101-element Vector{Float64}
  X: 3-element Vector{Float64}

julia> LAPACK.gglse!(ws, A, c, B, d)
([0.19723156207005318, 0.0683561362406917, 0.40981438442398854], 13.750943845251626)

LU

FastLapackInterface.LUWs — Type

LUWs

Workspace to be used with the LinearAlgebra.LU representation of the LU factorization which uses the LAPACK.getrf! function.

Examples

julia> A = [1.2 2.3
            6.2 3.3]
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

julia> ws = LUWs(A)
LUWs
  ipiv: 2-element Vector{Int64}

julia> t = LU(LAPACK.getrf!(ws, A)...)
LU{Float64, Matrix{Float64}, Vector{Int64}}
L factor:
2×2 Matrix{Float64}:
 1.0       0.0
 0.193548  1.0
U factor:
2×2 Matrix{Float64}:
 6.2  3.3
 0.0  1.66129

QR

FastLapackInterface.QRWs — Type

QRWs

Workspace for standard LinearAlgebra.QR factorization using the LAPACK.geqrf! function.

Examples

julia> A = [1.2 2.3
            6.2 3.3]
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

julia> ws = QRWs(A)
QRWs{Float64}
  work: 64-element Vector{Float64}
  τ: 2-element Vector{Float64}

julia> t = QR(LAPACK.geqrf!(ws, A)...)
QR{Float64, Matrix{Float64}, Vector{Float64}}
Q factor: 2×2 LinearAlgebra.QRPackedQ{Float64, Matrix{Float64}, Vector{Float64}}
R factor:
2×2 Matrix{Float64}:
 -6.31506  -3.67692
  0.0      -1.63102

julia> Matrix(t)
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

FastLapackInterface.QRWYWs — Type

QRWYWs

Workspace to be used with the LinearAlgebra.QRCompactWY representation of the blocked QR factorization which uses the LAPACK.geqrt! function. By default the blocksize for the algorithm is taken as min(36, min(size(template))), this can be overridden by using the blocksize keyword of the constructor.

Examples

julia> A = [1.2 2.3
            6.2 3.3]
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

julia> ws = QRWYWs(A)
QRWYWs{Float64, Matrix{Float64}}
  work: 4-element Vector{Float64}
  T: 2×2 Matrix{Float64}

julia> t = LinearAlgebra.QRCompactWY(LAPACK.geqrt!(ws, A)...)
LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}
Q factor: 2×2 LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}
R factor:
2×2 Matrix{Float64}:
 -6.31506  -3.67692
  0.0      -1.63102

julia> Matrix(t)
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

FastLapackInterface.QRPivotedWs — Type

QRPivotedWs

Workspace to be used with the LinearAlgebra.QRPivoted representation of the QR factorization which uses the LAPACK.geqp3! function.

Examples

julia> A = [1.2 2.3
            6.2 3.3]
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

julia> ws = QRPivotedWs(A)
QRPivotedWs{Float64, Float64}
  work: 100-element Vector{Float64}
  rwork: 0-element Vector{Float64}
  τ: 2-element Vector{Float64}
  jpvt: 2-element Vector{Int64}

julia> t = QRPivoted(LAPACK.geqp3!(ws, A)...)
QRPivoted{Float64, Matrix{Float64}, Vector{Float64}, Vector{Int64}}
Q factor: 2×2 LinearAlgebra.QRPackedQ{Float64, Matrix{Float64}, Vector{Float64}}
R factor:
2×2 Matrix{Float64}:
 -6.31506  -3.67692
  0.0      -1.63102
permutation:
2-element Vector{Int64}:
 1
 2

julia> Matrix(t)
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

FastLapackInterface.QROrmWs — Type

QROrmWs

Workspace to be used with the LinearAlgebra.LAPACK.ormqr! function. It requires the workspace of a QR or a QRPivoted previous factorization

Examples

```jldoctest julia> A = [1.2 2.3 6.2 3.3] 2×2 Matrix{Float64}: 1.2 2.3 6.2 3.3

julia> ws = QRPivotedWs(A) QRPivotedWs{Float64, Float64} work: 100-element Vector{Float64} rwork: 0-element Vector{Float64} τ: 2-element Vector{Float64} jpvt: 2-element Vector{Int64}

julia> C=[1 0.5; 2 1] 2×2 Matrix{Float64}: 1.0 0.5 2.0 1.0

julia> ormws = QROrmWs(ws, 'L', 'N', A, C) QROrmWs{Float64} work: 4224-element Vector{Float64} τ: 2-element Vector{Float64}

FastLapackInterface.QROrgWs — Type

QROrgWs

Workspace to be used with the LinearAlgebra.LAPACK.orgqr! function. It requires the workspace of a QR or a QRPivoted previous factorization

Examples

```jldoctest julia> A = [1.2 2.3 6.2 3.3] 2×2 Matrix{Float64}: 1.2 2.3 6.2 3.3

julia> ws = QRPivotedWs(A) QRPivotedWs{Float64, Float64} work: 100-element Vector{Float64} rwork: 0-element Vector{Float64} τ: 2-element Vector{Float64} jpvt: 2-element Vector{Int64}

julia> C=[1 0.5; 2 1] 2×2 Matrix{Float64}: 1.0 0.5 2.0 1.0

julia> orgws = QROrgWs(ws, 'L', 'N', A, C) QROrgWs{Float64} work: 4224-element Vector{Float64} τ: 2-element Vector{Float64}

Schur

FastLapackInterface.SchurWs — Type

SchurWs

Workspace to be used with the LinearAlgebra.Schur representation of the Schur decomposition which uses the LAPACK.gees! function.

Examples

julia> A = [1.2 2.3
            6.2 3.3]
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

julia> ws = SchurWs(A)
SchurWs{Float64}
  work: 68-element Vector{Float64}
  wr: 2-element Vector{Float64}
  wi: 2-element Vector{Float64}
  vs: 2×2 Matrix{Float64}
  sdim: Base.RefValue{Int64}
  bwork: 2-element Vector{Int64}
  eigen_values: 2-element Vector{ComplexF64}

julia> t = Schur(LAPACK.gees!(ws, 'V', A)...)
Schur{Float64, Matrix{Float64}, Vector{Float64}}
T factor:
2×2 Matrix{Float64}:
 -1.6695  -3.9
  0.0      6.1695
Z factor:
2×2 Matrix{Float64}:
 -0.625424  -0.780285
  0.780285  -0.625424
eigenvalues:
2-element Vector{Float64}:
 -1.6695025194532018
  6.169502519453203

julia> Matrix(t)
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

FastLapackInterface.GeneralizedSchurWs — Type

GeneralizedSchurWs

Workspace to be used with the LinearAlgebra.GeneralizedSchur representation of the Generalized Schur decomposition which uses the LAPACK.gges! function.

Examples

julia> A = [1.2 2.3
            6.2 3.3]
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

julia> B = [8.2 0.3
            1.7 4.3]
2×2 Matrix{Float64}:
 8.2  0.3
 1.7  4.3

julia> ws = GeneralizedSchurWs(A)
GeneralizedSchurWs{Float64}
  work: 90-element Vector{Float64}
  αr: 2-element Vector{Float64}
  αi: 2-element Vector{Float64}
  β: 2-element Vector{Float64}
  vsl: 2×2 Matrix{Float64}
  vsr: 2×2 Matrix{Float64}
  sdim: Base.RefValue{Int64}
  bwork: 2-element Vector{Int64}
  eigen_values: 2-element Vector{ComplexF64}
  
julia> t = GeneralizedSchur(LAPACK.gges!(ws, 'V','V', A, B)...)
GeneralizedSchur{Float64, Matrix{Float64}, Vector{ComplexF64}, Vector{Float64}}
S factor:
2×2 Matrix{Float64}:
 -1.43796  1.63843
  0.0      7.16295
T factor:
2×2 Matrix{Float64}:
 5.06887  -4.00221
 0.0       6.85558
Q factor:
2×2 Matrix{Float64}:
 -0.857329  0.514769
  0.514769  0.857329
Z factor:
2×2 Matrix{Float64}:
 -0.560266  0.828313
  0.828313  0.560266
α:
2-element Vector{ComplexF64}:
 -1.4379554610733563 + 0.0im
   7.162947865097022 + 0.0im
β:
2-element Vector{Float64}:
 5.068865029631368
 6.855578082442485

SVD

FastLapackInterface.SVDsddWs — Type

SVDsddWs

creates a workspace for LinearAlgebra.SVD factorization using the LAPACK.gesdd! function.

FastLapackInterface.SVDsddWs — Method

SVDsddWs(A; job = 'A')

creates a workspace for LinearAlgebra.SVD factorization using the LAPACK.gesdd! function.

Arguments

A::AbstractMatrix: the matrix to be decomposed. Only the shape of the matrix matters for building the workspace. The actual ellements value does not.
job::AbstractChar: one of A, N, O or S. Default = 'A'. See gesdd!.

Examples

julia> A = [1.2 2.3
            6.2 3.3]
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

julia> ws = FastLapackInterface.SVDsddWs(A, job = 'A')
SVDsddWs{Float64, Matrix{Float64}, Float64}
  U: 2×2 Matrix{Float64}
  VT: 2×2 Matrix{Float64}
  work: 134-element Vector{Float64}
  S: 2-element Vector{Float64}
  rwork: 0-element Vector{Float64}
  iwork: 16-element Vector{Int64}

julia> t = FastLapackInterface.gesdd!(ws, 'A', A);

julia> LinearAlgebra.SVD(t[1], t[2], t[3])
SVD{Float64, Float64, Matrix{Float64}, Vector{Float64}}
U factor:
2×2 Matrix{Float64}:
 -0.302437  -0.953169
 -0.953169   0.302437
singular values:
2-element Vector{Float64}:
 7.355199814161389
 1.400369841777625
Vt factor:
2×2 Matrix{Float64}:
 -0.852808  -0.522224
  0.522224  -0.852808

FastLapackInterface.SVDsvdWs — Type

SVDsvdWs

Workspace for LinearAlgebra.SVD factorization using the LAPACK.gesvd! function.

FastLapackInterface.SVDsvdWs — Method

SVDsvdWs(A; jobu = 'A', jobvt = 'A')

creates a workspace for LinearAlgebra.SVD factorization using the LAPACK.gesvd! function.

Arguments

A::AbstractMatrix: the matrix to be decomposed. Only the shape of the matrix matters for building the workspace. The actual ellements value does not.
jobu::AbstractChar: one of A, N, O or S. Default = 'A'. See gesvd!.
jobvt::AbstractChar: one of A, N, O or S. Default = 'A'. See gesvd!.

Examples

julia> A = [1.2 2.3
            6.2 3.3]
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

julia> ws = SVDsvdWs(A, jobu = 'A', jobvt = 'A')
SVDsvdWs{Float64, Matrix{Float64}, Float64}
  U: 2×2 Matrix{Float64}
  VT: 2×2 Matrix{Float64}
  work: 134-element Vector{Float64}
  S: 2-element Vector{Float64}
  rwork: 0-element Vector{Float64}

julia> t = FastLapackInterface.gesvd!(ws, 'A', 'A', A);

julia> SVD(t[1], t[2], t[3])
SVD{Float64, Float64, Matrix{Float64}, Vector{Float64}}
U factor:
2×2 Matrix{Float64}:
 -0.302437  -0.953169
 -0.953169   0.302437
singular values:
2-element Vector{Float64}:
 7.355199814161389
 1.400369841777625
Vt factor:
2×2 Matrix{Float64}:
 -0.852808  -0.522224
  0.522224  -0.852808

FastLapackInterface.GeneralizedSVDWs — Type

GeneralizedSVDWs

Workspace for LinearAlgebra.SVD factorization using the LAPACK.ggsvd3! function.

FastLapackInterface.GeneralizedSVDWs — Method

GeneralizedSVDWs(A::AbstractMatrix; jobu = 'U', jobv = 'V', jobq = 'Q')

creates a workspace for LinearAlgebra.SVD factorization using the LAPACK.ggsvd3! function.

Arguments

A::AbstractMatrix: the matrix to be decomposed. Only the shape of the matrix matters for building the workspace. The actual ellements value does not.
jobu::AbstractChar: one of U or N. Default = 'U'. See ggsvd3!.
jobv::AbstractChar: one of V or N. Default = 'V'. See ggsvd3!.
jobq::AbstractChar: one of Q or N. Default = 'Q'. See ggsvd3!.

Arguments

Examples

julia> A = [1.2 2.3
            6.2 3.3]
2×2 Matrix{Float64}:
 1.2  2.3
 6.2  3.3

julia> B = [2.2 3.3
            5.2 4.3]
2×2 Matrix{Float64}:
 2.2  3.3
 5.2  4.3

julia> ws = GeneralizedSVDWs(A, B)
GeneralizedSVDWs{Float64, Matrix{Float64}, Float64}
  alpha: 2-element Vector{Float64}
  beta: 2-element Vector{Float64}
  U: 2×2 Matrix{Float64}
  V: 2×2 Matrix{Float64}
  Q: 2×2 Matrix{Float64}
  work: 102-element Vector{Float64}
  rwork: 0-element Vector{Float64}
  iwork: 2-element Vector{Int64}


julia> t = FastLapackInterface.ggsvd3!(ws, 'U', 'V', 'Q', A, B);

julia> GeneralizedSVD(t...)
GeneralizedSVD{Float64, Matrix{Float64}, Float64, Vector{Float64}}
U factor:
2×2 Matrix{Float64}:
 -0.309806  0.9508
  0.9508    0.309806
V factor:
2×2 Matrix{Float64}:
 -0.653816  0.756653
  0.756653  0.653816
Q factor:
2×2 Matrix{Float64}:
  0.723532  0.690291
 -0.690291  0.723532
D1 factor:
2×2 Matrix{Float64}:
 0.911256  0.0
 0.0       0.517359
D2 factor:
2×2 Matrix{Float64}:
 0.411841  0.0
 0.0       0.855768
R0 factor:
2×2 Matrix{Float64}:
 2.54834  6.10941
 0.0      8.57328