Package Modelica.Math.Matrices.LAPACK
Interface to LAPACK library (should usually not directly be used but only indirectly via Modelica.Math.Matrices)

Information

This package contains external Modelica functions as interface to the LAPACK library (http://www.netlib.org/lapack) that provides FORTRAN subroutines to solve linear algebra tasks. Usually, these functions are not directly called, but only via the much more convenient interface of Modelica.Math.Matrices. The documentation of the LAPACK functions is a copy of the original FORTRAN code. The details of LAPACK are described in:

Anderson E., Bai Z., Bischof C., Blackford S., Demmel J., Dongarra J., Du Croz J., Greenbaum A., Hammarling S., McKenney A., and Sorensen D.:: Lapack Users' Guide. Third Edition, SIAM, 1999.

This package contains a direct interface to the LAPACK subroutines

Extends from Modelica.Icons.Package (Icon for standard packages).

Package Contents

Name	Description
`dgbsv`	Solve real system of linear equations A*X=B with a B matrix
`dgbsv_vec`	Solve real system of linear equations A*x=b with a b vector
`dgecon`	Estimates the reciprocal of the condition number of a general real matrix A
`dgees`	Computes real Schur form T of real nonsymmetric matrix A, and, optionally, the matrix of Schur vectors Z as well as the eigenvalues
`dgeev`	Compute eigenvalues and (right) eigenvectors for real nonsymmetric matrix A
`dgeev_eigenValues`	Compute eigenvalues for real nonsymmetric matrix A
`dgeevx`	Compute the eigenvalues and the (real) left and right eigenvectors of matrix A, using lapack routine dgeevx
`dgegv`	Compute generalized eigenvalues for a (A,B) system
`dgehrd`	reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q' * A * Q = H
`dgels_vec`	Solves overdetermined or underdetermined real linear equations A*x=b with a b vector
`dgelsx`	Computes the minimum-norm solution to a real linear least squares problem with rank deficient A
`dgelsx_vec`	Computes the minimum-norm solution to a real linear least squares problem with rank deficient A
`dgelsy`	Computes the minimum-norm solution to a real linear least squares problem with rank deficient A
`dgelsy_vec`	Computes the minimum-norm solution to a real linear least squares problem with rank deficient A
`dgeqpf`	Compute QR factorization of square or rectangular matrix A with column pivoting (A(:,p) = Q*R)
`dgeqrf`	computes a QR factorization without pivoting
`dgesdd`	Determine singular value decomposition
`dgesv`	Solve real system of linear equations A*X=B with a B matrix
`dgesv_vec`	Solve real system of linear equations A*x=b with a b vector
`dgesvd`	Determine singular value decomposition
`dgesvd_sigma`	Determine singular values
`dgesvx`	Solve real system of linear equations op(A)*X=B, op(A) is A or A' according to the Boolean input transposed
`dgetrf`	Compute LU factorization of square or rectangular matrix A (A = PLU)
`dgetri`	Computes the inverse of a matrix using the LU factorization from dgetrf(..)
`dgetrs`	Solves a system of linear equations with the LU decomposition from dgetrf(..)
`dgetrs_vec`	Solves a system of linear equations with the LU decomposition from dgetrf(..)
`dggev`	Compute generalized eigenvalues, as well as the left and right eigenvectors for a (A,B) system
`dggevx`	Compute generalized eigenvalues for a (A,B) system, using lapack routine dggevx
`dgglse_vec`	Solve a linear equality constrained least squares problem
`dgtsv`	Solve real system of linear equations A*X=B with B matrix and tridiagonal A
`dgtsv_vec`	Solve real system of linear equations A*x=b with b vector and tridiagonal A
`dhgeqz`	Compute generalized eigenvalues for a (A,B) system
`dhseqr`	Compute eigenvalues of a matrix H using lapack routine DHSEQR for Hessenberg form matrix
`dlange`	Norm of a matrix
`dorghr`	Generates a real orthogonal matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by DGEHRD
`dorgqr`	Generates a Real orthogonal matrix Q which is defined as the product of elementary reflectors as returned from dgeqpf
`dormhr`	overwrites the general real M-by-N matrix C with Q * C or C * Q or Q' * C or C * Q', where Q is an orthogonal matrix as returned by dgehrd
`dormqr`	overwrites the general real M-by-N matrix C with Q * C or C * Q or Q' * C or C * Q', where Q is an orthogonal matrix of a QR factorization as returned by dgeqrf
`dpotrf`	Computes the Cholesky factorization of a real symmetric positive definite matrix A
`dtrevc`	Compute the right and/or left eigenvectors of a real upper quasi-triangular matrix T
`dtrsen`	Reorder the real Schur factorization of a real matrix
`dtrsm`	Solve one of the matrix equations op( A )X = alphaB, or Xop( A ) = alphaB, where A is triangular matrix. BLAS routine
`dtrsyl`	Solve the real Sylvester matrix equation op(A)X + Xop(B) = scaleC or op(A)X - Xop(B) = scaleC

Function Modelica.Math.Matrices.LAPACK.dgeev
Compute eigenvalues and (right) eigenvectors for real nonsymmetric matrix A

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`

Outputs

Type	Name	Description
`Real`	`eigenReal[size(A, 1)]`	Real part of eigen values
`Real`	`eigenImag[size(A, 1)]`	Imaginary part of eigen values
`Real`	`eigenVectors[size(A, 1),size(A, 1)]`	Right eigen vectors
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgeev_eigenValues
Compute eigenvalues for real nonsymmetric matrix A

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`

Outputs

Type	Name	Description
`Real`	`EigenReal[size(A, 1)]`
`Real`	`EigenImag[size(A, 1)]`
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgegv
Compute generalized eigenvalues for a (A,B) system

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`
`Real`	`B[size(A, 1),size(A, 1)]`

Outputs

Type	Name	Description
`Real`	`alphaReal[size(A, 1)]`	Real part of alpha (eigenvalue=(alphaReal+i*alphaImag)/beta)
`Real`	`alphaImag[size(A, 1)]`	Imaginary part of alpha
`Real`	`beta[size(A, 1)]`	Denominator of eigenvalue
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgelsx
Computes the minimum-norm solution to a real linear least squares problem with rank deficient A

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`
`Real`	`B[size(A, 1),:]`
`Real`	`rcond`	Reciprocal condition number to estimate rank

Outputs

Type	Name	Description
`Real`	`X[max(size(A, 1), size(A, 2)),size(B, 2)]`	Solution is in first size(A,2) rows
`Integer`	`info`
`Integer`	`rank`	Effective rank of A

Function Modelica.Math.Matrices.LAPACK.dgelsx_vec
Computes the minimum-norm solution to a real linear least squares problem with rank deficient A

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`
`Real`	`b[size(A, 1)]`
`Real`	`rcond`	Reciprocal condition number to estimate rank

Outputs

Type	Name	Description
`Real`	`x[max(size(A, 1), size(A, 2))]`	solution is in first size(A,2) rows
`Integer`	`info`
`Integer`	`rank`	Effective rank of A

Function Modelica.Math.Matrices.LAPACK.dgelsy
Computes the minimum-norm solution to a real linear least squares problem with rank deficient A

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`
`Real`	`B[size(A, 1),:]`
`Real`	`rcond`	Reciprocal condition number to estimate rank

Outputs

Type	Name	Description
`Real`	`X[max(size(A, 1), size(A, 2)),size(B, 2)]`	Solution is in first size(A,2) rows
`Integer`	`info`
`Integer`	`rank`	Effective rank of A

Function Modelica.Math.Matrices.LAPACK.dgelsy_vec
Computes the minimum-norm solution to a real linear least squares problem with rank deficient A

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`
`Real`	`b[size(A, 1)]`
`Real`	`rcond`	Reciprocal condition number to estimate rank

Outputs

Type	Name	Description
`Real`	`x[max(size(A, 1), size(A, 2))]`	solution is in first size(A,2) rows
`Integer`	`info`
`Integer`	`rank`	Effective rank of A

Function Modelica.Math.Matrices.LAPACK.dgels_vec
Solves overdetermined or underdetermined real linear equations A*x=b with a b vector

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`
`Real`	`b[size(A, 1)]`

Outputs

Type	Name	Description
`Real`	`x[max(size(A, 1), size(A, 2))]`	solution is in first size(A,2) rows
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgesv
Solve real system of linear equations A*X=B with a B matrix

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`
`Real`	`B[size(A, 1),:]`

Outputs

Type	Name	Description
`Real`	`X[size(A, 1),size(B, 2)]`
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgesv_vec
Solve real system of linear equations A*x=b with a b vector

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`
`Real`	`b[size(A, 1)]`

Outputs

Type	Name	Description
`Real`	`x[size(A, 1)]`
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgglse_vec
Solve a linear equality constrained least squares problem

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Minimize \|A*x - c\|^2
`Real`	`c[size(A, 1)]`
`Real`	`B[:,size(A, 2)]`	subject to B*x=d
`Real`	`d[size(B, 1)]`

Outputs

Type	Name	Description
`Real`	`x[size(A, 2)]`	solution vector
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgtsv
Solve real system of linear equations A*X=B with B matrix and tridiagonal A

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`superdiag[:]`
`Real`	`diag[size(superdiag, 1) + 1]`
`Real`	`subdiag[size(superdiag, 1)]`
`Real`	`B[size(diag, 1),:]`

Outputs

Type	Name	Description
`Real`	`X[size(B, 1),size(B, 2)]`
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgtsv_vec
Solve real system of linear equations A*x=b with b vector and tridiagonal A

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`superdiag[:]`
`Real`	`diag[size(superdiag, 1) + 1]`
`Real`	`subdiag[size(superdiag, 1)]`
`Real`	`b[size(diag, 1)]`

Outputs

Type	Name	Description
`Real`	`x[size(b, 1)]`
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgbsv
Solve real system of linear equations A*X=B with a B matrix

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Integer`	`n`	Number of equations
`Integer`	`kLower`	Number of lower bands
`Integer`	`kUpper`	Number of upper bands
`Real`	`A[2 * kLower + kUpper + 1,n]`
`Real`	`B[n,:]`

Outputs

Type	Name	Description
`Real`	`X[n,size(B, 2)]`
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgbsv_vec
Solve real system of linear equations A*x=b with a b vector

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Integer`	`n`	Number of equations
`Integer`	`kLower`	Number of lower bands
`Integer`	`kUpper`	Number of upper bands
`Real`	`A[2 * kLower + kUpper + 1,n]`
`Real`	`b[n]`

Outputs

Type	Name	Description
`Real`	`x[n]`
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgesvd
Determine singular value decomposition

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`

Outputs

Type	Name	Description
`Real`	`sigma[min(size(A, 1), size(A, 2))]`
`Real`	`U[size(A, 1),size(A, 1)]`
`Real`	`VT[size(A, 2),size(A, 2)]`
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgesvd_sigma
Determine singular values

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`

Outputs

Type	Name	Description
`Real`	`sigma[min(size(A, 1), size(A, 2))]`
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgetrf
Compute LU factorization of square or rectangular matrix A (A = PLU)

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Square or rectangular matrix

Outputs

Type	Name	Description
`Real`	`LU[size(A, 1),size(A, 2)]`
`Integer`	`pivots[min(size(A, 1), size(A, 2))]`	Pivot vector
`Integer`	`info`	Information

Function Modelica.Math.Matrices.LAPACK.dgetrs
Solves a system of linear equations with the LU decomposition from dgetrf(..)

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`LU[:,size(LU, 1)]`	LU factorization of dgetrf of a square matrix
`Integer`	`pivots[size(LU, 1)]`	Pivot vector of dgetrf
`Real`	`B[size(LU, 1),:]`	Right hand side matrix B

Outputs

Type	Name	Description
`Real`	`X[size(B, 1),size(B, 2)]`	Solution matrix X
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgetrs_vec
Solves a system of linear equations with the LU decomposition from dgetrf(..)

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`LU[:,size(LU, 1)]`	LU factorization of dgetrf of a square matrix
`Integer`	`pivots[size(LU, 1)]`	Pivot vector of dgetrf
`Real`	`b[size(LU, 1)]`	Right hand side vector b

Outputs

Type	Name	Description
`Real`	`x[size(b, 1)]`
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgetri
Computes the inverse of a matrix using the LU factorization from dgetrf(..)

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`LU[:,size(LU, 1)]`	LU factorization of dgetrf of a square matrix
`Integer`	`pivots[size(LU, 1)]`	Pivot vector of dgetrf

Outputs

Type	Name	Description
`Real`	`inv[size(LU, 1),size(LU, 2)]`	Inverse of matrix PLU
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgeqpf
Compute QR factorization of square or rectangular matrix A with column pivoting (A(:,p) = Q*R)

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Square or rectangular matrix

Outputs

Type	Name	Description
`Real`	`QR[size(A, 1),size(A, 2)]`	QR factorization in packed format
`Real`	`tau[min(size(A, 1), size(A, 2))]`	The scalar factors of the elementary reflectors of Q
`Integer`	`p[size(A, 2)]`	Pivot vector
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dorgqr
Generates a Real orthogonal matrix Q which is defined as the product of elementary reflectors as returned from dgeqpf

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`QR[:,:]`	QR from dgeqpf
`Real`	`tau[min(size(QR, 1), size(QR, 2))]`	The scalar factors of the elementary reflectors of Q

Outputs

Type	Name	Description
`Real`	`Q[size(QR, 1),size(QR, 2)]`	Orthogonal matrix Q
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgees
Computes real Schur form T of real nonsymmetric matrix A, and, optionally, the matrix of Schur vectors Z as well as the eigenvalues

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Square matrix

Outputs

Type	Name	Description
`Real`	`T[size(A, 1),size(A, 2)]`	Real Schur form with A = ZTZ'
`Real`	`Z[size(A, 1),size(A, 1)]`	orthogonal matrix Z of Schur vectors
`Real`	`eval_real[size(A, 1)]`	real part of the eigenvectors of A
`Real`	`eval_imag[size(A, 1)]`	imaginary part of the eigenvectors of A
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dtrsen
Reorder the real Schur factorization of a real matrix

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`String`	`job`	Specifies the usage of a condition number
`String`	`compq`	Is "V" if Schur vector matrix is to be updated
`Boolean`	`select[:]`	Specifies the eigenvalues to reorder
`Real`	`T[:,:]`	Real Schur form to be reordered
`Real`	`Q[:,size(T, 2)]`	Matrix of the Schur vectors

Outputs

Type	Name	Description
`Real`	`To[:,:]`	Reordered Schur form
`Real`	`Qo[:,:]`	Reordered Schur vectors
`Real`	`wr[size(T, 2)]`	Reordered eigenvalues, real part
`Real`	`wi[size(T, 2)]`	Reordered eigenvalues, imaginary part
`Integer`	`m`	Dimension of the invariant sub space spanned bei the selected eigenvalues
`Real`	`s`	Lower bound of the reciprocal condition number. Not referenced for job==V
`Real`	`sep`	Estimated reciprocal condition number of the specified invariant subspace
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgesvx
Solve real system of linear equations op(A)*X=B, op(A) is A or A' according to the Boolean input transposed

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Real square matrix A
`Real`	`B[size(A, 1),:]`	Real matrix B
`Boolean`	`transposed`	True if the equation to be solved is A'*X=B

Outputs

Type	Name	Description
`Real`	`X[size(A, 1),size(B, 2)]`	Solution matrix
`Integer`	`info`
`Real`	`rcond`	reciprocal condition number of the matrix A

Function Modelica.Math.Matrices.LAPACK.dtrsyl
Solve the real Sylvester matrix equation op(A)X + Xop(B) = scaleC or op(A)X - Xop(B) = scaleC

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Upper quais-triangular matrix
`Real`	`B[:,:]`	Upper quais-triangular matrix
`Real`	`C[if tranA then size(A, 1) else size(A, 2),if tranB then size(B, 1) else size(B, 2)]`	Right side of the Sylvester equation
`Boolean`	`tranA`	True if op(A)=A'
`Boolean`	`tranB`	True if op(B)=B'
`Integer`	`isgn`	Specifies the sign in the equation, +1 or -1

Outputs

Type	Name	Description
`Real`	`X[size(C, 1),size(C, 2)]`	Solution of the Sylvester equation
`Real`	`scale`	Scale factor
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dhseqr
Compute eigenvalues of a matrix H using lapack routine DHSEQR for Hessenberg form matrix

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`H[:,size(H, 1)]`	Matrix H with Hessenberg form
`Boolean`	`eigenValuesOnly`	True to compute the eigenvalues. False to compute the Schur form too
`String`	`compz`	Specifies the computation of the Schur vectors
`Real`	`Z[:,:]`	Matrix Z

Outputs

Type	Name	Description
`Real`	`alphaReal[size(H, 1)]`	Real part of alpha (eigenvalue=(alphaReal+i*alphaImag))
`Real`	`alphaImag[size(H, 1)]`	Imaginary part of alpha (eigenvalue=(alphaReal+i*alphaImag))
`Integer`	`info`
`Real`	`Ho[:,:]`	Schur decomposition (if eigenValuesOnly==false, unspecified else)
`Real`	`Zo[:,:]`
`Real`	`work[3 * max(1, size(H, 1))]`

Function Modelica.Math.Matrices.LAPACK.dlange
Norm of a matrix

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Real matrix A
`String`	`norm`	specifies the norm, i.e., 1, I, F, M

Outputs

Type	Name	Description
`Real`	`anorm`	norm of A

Function Modelica.Math.Matrices.LAPACK.dgecon
Estimates the reciprocal of the condition number of a general real matrix A

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`LU_of_A[:,:]`	LU factorization of a real matrix A
`Boolean`	`inf`	Is true if infinity norm is used and false for 1-norm
`Real`	`anorm`	norm of A

Outputs

Type	Name	Description
`Real`	`rcond`	Reciprocal condition number of A
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgehrd
reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q' * A * Q = H

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`
`Integer`	`ilo`	lowest index where the original matrix had been Hessenbergform
`Integer`	`ihi`	highest index where the original matrix had been Hessenbergform

Outputs

Type	Name	Description
`Real`	`Aout[size(A, 1),size(A, 2)]`	contains the Hessenberg form in the upper triangle and the first subdiagonal and below the first subdiagonal it contains the elementary reflectors which represents (with array tau) as a product the orthogonal matrix Q
`Real`	`tau[max(size(A, 1), 1) - 1]`	scalar factors of the elementary reflectors
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgeqrf
computes a QR factorization without pivoting

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Square or rectangular matrix

Outputs

Type	Name	Description
`Real`	`Aout[size(A, 1),size(A, 2)]`	the upper triangle of the array contains the upper trapezoidal matrix R; the elements below the diagonal, together with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors
`Real`	`tau[min(size(A, 1), size(A, 2))]`	scalar factors of the elementary reflectors
`Integer`	`info`
`Real`	`work[3 * max(1, size(A, 2))]`

Function Modelica.Math.Matrices.LAPACK.dgeevx
Compute the eigenvalues and the (real) left and right eigenvectors of matrix A, using lapack routine dgeevx

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`

Outputs

Type	Name	Description
`Real`	`alphaReal[size(A, 1)]`	Real part of alpha (eigenvalue=(alphaReal+i*alphaImag))
`Real`	`alphaImag[size(A, 1)]`	Imaginary part of alpha (eigenvalue=(alphaReal+i*alphaImag))
`Real`	`lEigenVectors[size(A, 1),size(A, 1)]`	left eigenvectors of matrix A
`Real`	`rEigenVectors[size(A, 1),size(A, 1)]`	right eigenvectors of matrix A
`Real`	`AS[size(A, 1),size(A, 2)]`	AS iss the real Schur form of the balanced version of the input matrix A
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dgesdd
Determine singular value decomposition

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`

Outputs

Type	Name	Description
`Real`	`sigma[min(size(A, 1), size(A, 2))]`
`Real`	`U[size(A, 1),size(A, 1)]`
`Real`	`VT[size(A, 2),size(A, 2)]`
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dggev
Compute generalized eigenvalues, as well as the left and right eigenvectors for a (A,B) system

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`
`Real`	`B[size(A, 1),size(A, 1)]`
`Integer`	`nA`	The actual dimensions of matrices A and B (the computation is performed for A[1:nA,1:nA], B[1:nA,1:nA])

Outputs

Type	Name	Description
`Real`	`alphaReal[size(A, 1)]`	Real part of alpha (eigenvalue=(alphaReal+i*alphaImag)/beta)
`Real`	`alphaImag[size(A, 1)]`	Imaginary part of alpha
`Real`	`beta[size(A, 1)]`	Denominator of eigenvalue
`Real`	`lEigenVectors[size(A, 1),size(A, 1)]`	left eigenvectors of matrix A
`Real`	`rEigenVectors[size(A, 1),size(A, 1)]`	right eigenvectors of matrix A
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dggevx
Compute generalized eigenvalues for a (A,B) system, using lapack routine dggevx

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`
`Real`	`B[size(A, 1),size(A, 1)]`

Outputs

Type	Name	Description
`Real`	`alphaReal[size(A, 1)]`	Real part of alpha (eigenvalue=(alphaReal+i*alphaImag)/beta)
`Real`	`alphaImag[size(A, 1)]`	Imaginary part of alpha
`Real`	`beta[size(A, 1)]`	Denominator of eigenvalue
`Real`	`lEigenVectors[size(A, 1),size(A, 1)]`	left eigenvectors of matrix A
`Real`	`rEigenVectors[size(A, 1),size(A, 1)]`	right eigenvectors of matrix A
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dhgeqz
Compute generalized eigenvalues for a (A,B) system

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`
`Real`	`B[size(A, 1),size(A, 1)]`

Outputs

Type	Name	Description
`Real`	`alphaReal[size(A, 1)]`	Real part of alpha (eigenvalue=(alphaReal+i*alphaImag)/beta)
`Real`	`alphaImag[size(A, 1)]`	Imaginary part of alpha
`Real`	`beta[size(A, 1)]`	Denominator of eigenvalue
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dormhr
overwrites the general real M-by-N matrix C with Q * C or C * Q or Q' * C or C * Q', where Q is an orthogonal matrix as returned by dgehrd

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`C[:,:]`
`Real`	`A[:,:]`
`Real`	`tau[if side == "L" then size(C, 2) - 1 else size(C, 1) - 1]`
`String`	`side`
`String`	`trans`
`Integer`	`ilo`	lowest index where the original matrix had been Hessenbergform
`Integer`	`ihi`	highest index where the original matrix had been Hessenbergform

Outputs

Type	Name	Description
`Real`	`Cout[size(C, 1),size(C, 2)]`	contains the Hessenberg form in the upper triangle and the first subdiagonal and below the first subdiagonal it contains the elementary reflectors which represents (with array tau) as a product the orthogonal matrix Q
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dormqr
overwrites the general real M-by-N matrix C with Q * C or C * Q or Q' * C or C * Q', where Q is an orthogonal matrix of a QR factorization as returned by dgeqrf

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`C[:,:]`
`Real`	`A[:,:]`
`Real`	`tau[:]`
`String`	`side`
`String`	`trans`

Outputs

Type	Name	Description
`Real`	`Cout[size(C, 1),size(C, 2)]`	contains QC or QTC or CQT or CQ
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dtrevc
Compute the right and/or left eigenvectors of a real upper quasi-triangular matrix T

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`T[:,size(T, 1)]`	Upper quasi triangular matrix
`String`	`side`	Specify which eigenvectors
`String`	`howmny`	Specify how many eigenvectors
`Real`	`Q[size(T, 1),size(T, 1)]`	Orthogonal matrix Q of Schur vectors returned by DHSEQR

Outputs

Type	Name	Description
`Real`	`lEigenVectors[size(T, 1),size(T, 1)]`	left eigenvectors of matrix T
`Real`	`rEigenVectors[size(T, 1),size(T, 1)]`	right eigenvectors of matrix T
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dpotrf
Computes the Cholesky factorization of a real symmetric positive definite matrix A

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Real symmetric positive definite matrix A
`Boolean`	`upper`	True if the upper triangle of A is provided

Outputs

Type	Name	Description
`Real`	`Acholesky[size(A, 1),size(A, 1)]`	Cholesky factor
`Integer`	`info`

Function Modelica.Math.Matrices.LAPACK.dtrsm
Solve one of the matrix equations op( A )X = alphaB, or Xop( A ) = alphaB, where A is triangular matrix. BLAS routine

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,:]`	Input matrix A
`Real`	`B[:,:]`	Input matrix B
`Real`	`alpha`	Factor alpha
`Boolean`	`right`	True if A is right multiplication
`Boolean`	`upper`	True if A is upper triangular
`Boolean`	`trans`	True if op(A) means transposed(A)
`Boolean`	`unitTriangular`	True if A is unit triangular, i.e., all diagonal elements of A are equal to 1

Outputs

Type	Name	Description
`Real`	`X[size(B, 1),size(B, 2)]`	Matrix Bout=alphaop( A )B, or B := alphaBop( A )

Function Modelica.Math.Matrices.LAPACK.dorghr
Generates a real orthogonal matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by DGEHRD

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Type	Name	Description
`Real`	`A[:,size(A, 1)]`	Square matrix with the elementary reflectors
`Integer`	`ilo`	lowest index where the original matrix had been Hessenbergform - ilo must have the same value as in the previous call of DGEHRD
`Integer`	`ihi`	highest index where the original matrix had been Hessenbergform - ihi must have the same value as in the previous call of DGEHRD
`Real`	`tau[max(0, size(A, 1) - 1)]`	scalar factors of the elementary reflectors

Outputs

Type	Name	Description
`Real`	`Aout[size(A, 1),size(A, 2)]`	Orthogonal matrix as a result of elementary reflectors
`Integer`	`info`

Package Modelica.​Math.​Matrices.​LAPACKInterface to LAPACK library (should usually not directly be used but only indirectly via Modelica.Math.Matrices)

Information

Package Contents

Function Modelica.​Math.​Matrices.​LAPACK.​dgeevCompute eigenvalues and (right) eigenvectors for real nonsymmetric matrix A

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgeev_eigenValuesCompute eigenvalues for real nonsymmetric matrix A

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgegvCompute generalized eigenvalues for a (A,B) system

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgelsxComputes the minimum-norm solution to a real linear least squares problem with rank deficient A

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgelsx_vecComputes the minimum-norm solution to a real linear least squares problem with rank deficient A

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgelsyComputes the minimum-norm solution to a real linear least squares problem with rank deficient A

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgelsy_vecComputes the minimum-norm solution to a real linear least squares problem with rank deficient A

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgels_vecSolves overdetermined or underdetermined real linear equations A*x=b with a b vector

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgesvSolve real system of linear equations A*X=B with a B matrix

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgesv_vecSolve real system of linear equations A*x=b with a b vector

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgglse_vecSolve a linear equality constrained least squares problem

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgtsvSolve real system of linear equations A*X=B with B matrix and tridiagonal A

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgtsv_vecSolve real system of linear equations A*x=b with b vector and tridiagonal A

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgbsvSolve real system of linear equations A*X=B with a B matrix

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgbsv_vecSolve real system of linear equations A*x=b with a b vector

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgesvdDetermine singular value decomposition

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgesvd_sigmaDetermine singular values

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgetrfCompute LU factorization of square or rectangular matrix A (A = P*L*U)

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgetrsSolves a system of linear equations with the LU decomposition from dgetrf(..)

Information

Inputs

Outputs

Function Modelica.​Math.​Matrices.​LAPACK.​dgetrs_vecSolves a system of linear equations with the LU decomposition from dgetrf(..)

Package Modelica.Math.Matrices.LAPACK
Interface to LAPACK library (should usually not directly be used but only indirectly via Modelica.Math.Matrices)

Function Modelica.Math.Matrices.LAPACK.dgeev
Compute eigenvalues and (right) eigenvectors for real nonsymmetric matrix A

Function Modelica.Math.Matrices.LAPACK.dgeev_eigenValues
Compute eigenvalues for real nonsymmetric matrix A

Function Modelica.Math.Matrices.LAPACK.dgegv
Compute generalized eigenvalues for a (A,B) system

Function Modelica.Math.Matrices.LAPACK.dgelsx
Computes the minimum-norm solution to a real linear least squares problem with rank deficient A

Function Modelica.Math.Matrices.LAPACK.dgelsx_vec
Computes the minimum-norm solution to a real linear least squares problem with rank deficient A

Function Modelica.Math.Matrices.LAPACK.dgelsy
Computes the minimum-norm solution to a real linear least squares problem with rank deficient A

Function Modelica.Math.Matrices.LAPACK.dgelsy_vec
Computes the minimum-norm solution to a real linear least squares problem with rank deficient A

Function Modelica.Math.Matrices.LAPACK.dgels_vec
Solves overdetermined or underdetermined real linear equations A*x=b with a b vector

Function Modelica.Math.Matrices.LAPACK.dgesv
Solve real system of linear equations A*X=B with a B matrix

Function Modelica.Math.Matrices.LAPACK.dgesv_vec
Solve real system of linear equations A*x=b with a b vector

Function Modelica.Math.Matrices.LAPACK.dgglse_vec
Solve a linear equality constrained least squares problem

Function Modelica.Math.Matrices.LAPACK.dgtsv
Solve real system of linear equations A*X=B with B matrix and tridiagonal A

Function Modelica.Math.Matrices.LAPACK.dgtsv_vec
Solve real system of linear equations A*x=b with b vector and tridiagonal A

Function Modelica.Math.Matrices.LAPACK.dgbsv
Solve real system of linear equations A*X=B with a B matrix

Function Modelica.Math.Matrices.LAPACK.dgbsv_vec
Solve real system of linear equations A*x=b with a b vector

Function Modelica.Math.Matrices.LAPACK.dgesvd
Determine singular value decomposition

Function Modelica.Math.Matrices.LAPACK.dgesvd_sigma
Determine singular values

Function Modelica.Math.Matrices.LAPACK.dgetrf
Compute LU factorization of square or rectangular matrix A (A = PLU)

Function Modelica.Math.Matrices.LAPACK.dgetrs
Solves a system of linear equations with the LU decomposition from dgetrf(..)

Function Modelica.Math.Matrices.LAPACK.dgetrs_vec
Solves a system of linear equations with the LU decomposition from dgetrf(..)

Function Modelica.Math.Matrices.LAPACK.dgetri
Computes the inverse of a matrix using the LU factorization from dgetrf(..)

Function Modelica.Math.Matrices.LAPACK.dgeqpf
Compute QR factorization of square or rectangular matrix A with column pivoting (A(:,p) = Q*R)

Function Modelica.Math.Matrices.LAPACK.dorgqr
Generates a Real orthogonal matrix Q which is defined as the product of elementary reflectors as returned from dgeqpf

Function Modelica.Math.Matrices.LAPACK.dgees
Computes real Schur form T of real nonsymmetric matrix A, and, optionally, the matrix of Schur vectors Z as well as the eigenvalues

Function Modelica.Math.Matrices.LAPACK.dtrsen
Reorder the real Schur factorization of a real matrix

Function Modelica.Math.Matrices.LAPACK.dgesvx
Solve real system of linear equations op(A)*X=B, op(A) is A or A' according to the Boolean input transposed

Function Modelica.Math.Matrices.LAPACK.dtrsyl
Solve the real Sylvester matrix equation op(A)X + Xop(B) = scaleC or op(A)X - Xop(B) = scaleC

Function Modelica.Math.Matrices.LAPACK.dhseqr
Compute eigenvalues of a matrix H using lapack routine DHSEQR for Hessenberg form matrix

Function Modelica.Math.Matrices.LAPACK.dlange
Norm of a matrix

Function Modelica.Math.Matrices.LAPACK.dgecon
Estimates the reciprocal of the condition number of a general real matrix A

Function Modelica.Math.Matrices.LAPACK.dgehrd
reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q' * A * Q = H

Function Modelica.Math.Matrices.LAPACK.dgeqrf
computes a QR factorization without pivoting

Function Modelica.Math.Matrices.LAPACK.dgeevx
Compute the eigenvalues and the (real) left and right eigenvectors of matrix A, using lapack routine dgeevx

Function Modelica.Math.Matrices.LAPACK.dgesdd
Determine singular value decomposition

Function Modelica.Math.Matrices.LAPACK.dggev
Compute generalized eigenvalues, as well as the left and right eigenvectors for a (A,B) system

Function Modelica.Math.Matrices.LAPACK.dggevx
Compute generalized eigenvalues for a (A,B) system, using lapack routine dggevx

Function Modelica.Math.Matrices.LAPACK.dhgeqz
Compute generalized eigenvalues for a (A,B) system