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Octopus
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Octopus
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Data Types | |
| interface | lalg_cholesky |
| interface | lalg_determinant |
| Note that lalg_determinant and lalg_inverse are just wrappers over the same routine. More... | |
| interface | lalg_eigensolve |
| interface | lalg_eigensolve_nonh |
| interface | lalg_eigensolve_parallel |
| interface | lalg_geneigensolve |
| interface | lalg_inverse |
| interface | lalg_least_squares |
| interface | lalg_linsyssolve |
| interface | lalg_lowest_eigensolve |
| interface | lalg_lowest_geneigensolve |
| interface | lalg_matrix_function |
| interface | lalg_matrix_norm2 |
| interface | lalg_pseudo_inverse |
| interface | lalg_singular_value_decomp |
| interface | lalg_svd_inverse |
| interface | lapack_geev |
Functions/Subroutines | |
| real(real64) function | sfmin () |
| Auxiliary function. More... | |
| subroutine | lalg_dgeev (jobvl, jobvr, n, a, lda, w, vl, ldvl, vr, ldvr, work, lwork, rwork, info) |
| subroutine | lalg_zgeev (jobvl, jobvr, n, a, lda, w, vl, ldvl, vr, ldvr, work, lwork, rwork, info) |
| subroutine, public | zlalg_exp (nn, pp, aa, ex, hermitian) |
| subroutine, public | zlalg_phi (nn, pp, aa, ex, hermitian) |
| subroutine, public | lalg_zeigenderivatives (n, mat, zeigenvec, zeigenval, zmat) |
| subroutine | lalg_zpseudoinverse (n, mat, imat) |
| Computes the Moore-Penrose pseudoinverse of a complex matrix. More... | |
| subroutine, public | lalg_check_zeigenderivatives (n, mat) |
| complex(real64) function, public | lalg_zdni (eigenvec, alpha, beta) |
| complex(real64) function, public | lalg_zduialpha (eigenvec, mmatrix, alpha, gamma, delta) |
| complex(real64) function, public | lalg_zd2ni (eigenvec, mmatrix, alpha, beta, gamma, delta) |
| pure real(real64) function | pseudoinverse_default_tolerance (m, n, sg_values) |
| Computes the default Moore-Penrose pseudoinverse tolerance for zeroing. More... | |
| real(real64) function, dimension(1:n, 1:n), public | lalg_remove_rotation (n, A) |
| Remove rotation from affine transformation A by computing the polar decomposition and discarding the rotational part. The polar decomposition of A is given by A = U P with P = sqrt(A^T A), where U is a rotation matrix and P is a scaling matrix. This function returns P. More... | |
| subroutine | zcholesky (n, a, bof, err_code) |
| Compute the Cholesky decomposition of real symmetric or complex Hermitian positive definite matrix a, dim(a) = n x n. On return a = u^T u with u upper triangular matrix. More... | |
| subroutine | zgeneigensolve (n, a, b, e, preserve_mat, bof, err_code) |
| Computes all the eigenvalues and the eigenvectors of a real symmetric or complex Hermitian generalized definite eigenproblem, of the form \( Ax=\lambda Bx \). B is also positive definite. More... | |
| subroutine | zeigensolve_nonh (n, a, e, err_code, side, sort_eigenvectors) |
| Computes all the eigenvalues and the right (left) eigenvectors of a real or complex (non-Hermitian) eigenproblem, of the form A*x=(lambda)*x. More... | |
| subroutine | zlowest_geneigensolve (k, n, a, b, e, v, preserve_mat, bof, err_code) |
| Computes the k lowest eigenvalues and the eigenvectors of a real symmetric or complex Hermitian generalized definite eigenproblem, of the form A*x=(lambda)*B*x. B is also positive definite. More... | |
| subroutine | zeigensolve (n, a, e, bof, err_code) |
| Computes all eigenvalues and eigenvectors of a real symmetric or hermitian square matrix A. More... | |
| subroutine | zlowest_eigensolve (k, n, a, e, v, preserve_mat) |
| Computes the k lowest eigenvalues and the eigenvectors of a standard symmetric-definite eigenproblem, of the form A*x=(lambda)*x. Here A is assumed to be symmetric. More... | |
| complex(real64) function | zdeterminant (n, a, preserve_mat) |
| Invert a real symmetric or complex Hermitian square matrix a. More... | |
| subroutine | zdirect_inverse (n, a, det) |
| Invert a real symmetric or complex Hermitian square matrix a. More... | |
| subroutine | zmatrix_norm2 (m, n, a, norm_l2, preserve_mat) |
| Norm of a 2D matrix. More... | |
| subroutine | zsym_inverse (uplo, n, a) |
| Invert a real/complex symmetric square matrix a. More... | |
| subroutine | zlinsyssolve (n, nrhs, a, b, x) |
| compute the solution to a complex system of linear equations A*X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices. More... | |
| subroutine | zsingular_value_decomp (m, n, a, u, vt, sg_values, preserve_mat) |
| Computes the singular value decomposition of a complex MxN matrix a. More... | |
| subroutine | zsvd_inverse (m, n, a, threshold) |
| Computes inverse of a complex MxN matrix, a, using the SVD decomposition. More... | |
| subroutine | zlalg_pseudo_inverse (a, threshold) |
| Invert a matrix with the Moore-Penrose pseudo-inverse. More... | |
| subroutine | zupper_triangular_inverse (n, a) |
| Calculate the inverse of a real/complex upper triangular matrix (in unpacked storage). (lower triangular would be a trivial variant of this) More... | |
| subroutine | zleast_squares_vec (nn, aa, bb, xx, preserve_mat) |
| subroutine | zeigensolve_parallel (n, a, e, bof, err_code) |
| Computes all the eigenvalues and the eigenvectors of a real symmetric or complex Hermitian eigenproblem in parallel using ScaLAPACK or ELPA on all processors n: dimension of matrix a: input matrix, on exit: contains eigenvectors e: eigenvalues. More... | |
| subroutine | zinverse (n, a, method, det, threshold, uplo) |
| An interface to different method to invert a matrix. More... | |
| subroutine, public | zlalg_matrix_function (n, factor, a, fun_a, fun, hermitian) |
| This routine calculates a function of a matrix by using an eigenvalue decomposition. More... | |
| subroutine | dcholesky (n, a, bof, err_code) |
| Compute the Cholesky decomposition of real symmetric or complex Hermitian positive definite matrix a, dim(a) = n x n. On return a = u^T u with u upper triangular matrix. More... | |
| subroutine | dgeneigensolve (n, a, b, e, preserve_mat, bof, err_code) |
| Computes all the eigenvalues and the eigenvectors of a real symmetric or complex Hermitian generalized definite eigenproblem, of the form \( Ax=\lambda Bx \). B is also positive definite. More... | |
| subroutine | deigensolve_nonh (n, a, e, err_code, side, sort_eigenvectors) |
| Computes all the eigenvalues and the right (left) eigenvectors of a real or complex (non-Hermitian) eigenproblem, of the form A*x=(lambda)*x. More... | |
| subroutine | dlowest_geneigensolve (k, n, a, b, e, v, preserve_mat, bof, err_code) |
| Computes the k lowest eigenvalues and the eigenvectors of a real symmetric or complex Hermitian generalized definite eigenproblem, of the form A*x=(lambda)*B*x. B is also positive definite. More... | |
| subroutine | deigensolve (n, a, e, bof, err_code) |
| Computes all eigenvalues and eigenvectors of a real symmetric or hermitian square matrix A. More... | |
| subroutine | dlowest_eigensolve (k, n, a, e, v, preserve_mat) |
| Computes the k lowest eigenvalues and the eigenvectors of a standard symmetric-definite eigenproblem, of the form A*x=(lambda)*x. Here A is assumed to be symmetric. More... | |
| real(real64) function | ddeterminant (n, a, preserve_mat) |
| Invert a real symmetric or complex Hermitian square matrix a. More... | |
| subroutine | ddirect_inverse (n, a, det) |
| Invert a real symmetric or complex Hermitian square matrix a. More... | |
| subroutine | dmatrix_norm2 (m, n, a, norm_l2, preserve_mat) |
| Norm of a 2D matrix. More... | |
| subroutine | dsym_inverse (uplo, n, a) |
| Invert a real/complex symmetric square matrix a. More... | |
| subroutine | dlinsyssolve (n, nrhs, a, b, x) |
| compute the solution to a real system of linear equations A*X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices. More... | |
| subroutine | dsingular_value_decomp (m, n, a, u, vt, sg_values, preserve_mat) |
| Computes the singular value decomposition of a real M x N matrix a. More... | |
| subroutine | dsvd_inverse (m, n, a, threshold) |
| Computes the inverse of a real M x N matrix, a, using the SVD decomposition. More... | |
| subroutine | dlalg_pseudo_inverse (a, threshold) |
| Invert a matrix with the Moore-Penrose pseudo-inverse. More... | |
| subroutine | dupper_triangular_inverse (n, a) |
| Calculate the inverse of a real/complex upper triangular matrix (in unpacked storage). (lower triangular would be a trivial variant of this) More... | |
| subroutine | dleast_squares_vec (nn, aa, bb, xx, preserve_mat) |
| subroutine | deigensolve_parallel (n, a, e, bof, err_code) |
| Computes all the eigenvalues and the eigenvectors of a real symmetric or complex Hermitian eigenproblem in parallel using ScaLAPACK or ELPA on all processors n: dimension of matrix a: input matrix, on exit: contains eigenvectors e: eigenvalues. More... | |
| subroutine | dinverse (n, a, method, det, threshold, uplo) |
| An interface to different method to invert a matrix. More... | |
| subroutine | dlalg_matrix_function (n, factor, a, fun_a, fun, hermitian) |
| This routine calculates a function of a matrix by using an eigenvalue decomposition. More... | |
|
private |
Auxiliary function.
Definition at line 240 of file lalg_adv.F90.
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private |
| [in,out] | a | a(lda,n) |
| [out] | w | w(n) |
| [out] | vr | vl(ldvl,n), vl(ldvr,n) |
| [out] | rwork | rwork(max(1,2n)) |
| [out] | work | work(lwork) |
Definition at line 252 of file lalg_adv.F90.
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private |
| [in,out] | a | a(lda,n) |
| [out] | w | w(n) |
| [out] | vr | vl(ldvl,n), vl(ldvr,n) |
| [out] | rwork | rwork(max(1,2n)) |
| [out] | work | work(lwork) |
Definition at line 279 of file lalg_adv.F90.
| subroutine, public lalg_adv_oct_m::zlalg_exp | ( | integer, intent(in) | nn, |
| complex(real64), intent(in) | pp, | ||
| complex(real64), dimension(:, :), intent(in) | aa, | ||
| complex(real64), dimension(:, :), intent(inout) | ex, | ||
| logical, intent(in) | hermitian | ||
| ) |
This routine calculates the exponential of a matrix by using an eigenvalue decomposition.
For the hermitian case:
A = V D V^T => exp(A) = V exp(D) V^T
and in general
A = V D V^-1 => exp(A) = V exp(D) V^-1
This is slow but it is simple to implement, and for the moment it does not affect performance.
Definition at line 313 of file lalg_adv.F90.
| subroutine, public lalg_adv_oct_m::zlalg_phi | ( | integer, intent(in) | nn, |
| complex(real64), intent(in) | pp, | ||
| complex(real64), dimension(:, :), intent(in) | aa, | ||
| complex(real64), dimension(:, :), intent(inout) | ex, | ||
| logical, intent(in) | hermitian | ||
| ) |
This routine calculates phi(pp*A), where A is a matrix, pp is any complex number, and phi is the function:
phi(x) = (e^x - 1)/x
For the Hermitian case, for any function f:
A = V D V^T => f(A) = V f(D) V^T
and in general
A = V D V^-1 => f(A) = V f(D) V^-1
Definition at line 391 of file lalg_adv.F90.
| subroutine, public lalg_adv_oct_m::lalg_zeigenderivatives | ( | integer, intent(in) | n, |
| complex(real64), dimension(:, :), intent(in), contiguous | mat, | ||
| complex(real64), dimension(:, :), intent(out), contiguous | zeigenvec, | ||
| complex(real64), dimension(:), intent(out), contiguous | zeigenval, | ||
| complex(real64), dimension(:, :, :), intent(out), contiguous | zmat | ||
| ) |
Computes the necessary ingredients to obtain, later, the first and second derivatives of the eigenvalues of a Hermitean complex matrix zmat, and the first derivatives of the eigenvectors.
This follows the scheme of J. R. Magnus, Econometric Theory 1, 179 (1985), restricted to Hermitean matrices, although probably this can be
Definition at line 466 of file lalg_adv.F90.
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private |
Computes the Moore-Penrose pseudoinverse of a complex matrix.
Definition at line 518 of file lalg_adv.F90.
| subroutine, public lalg_adv_oct_m::lalg_check_zeigenderivatives | ( | integer, intent(in) | n, |
| complex(real64), dimension(:, :), intent(in) | mat | ||
| ) |
The purpose of this routine is to check that "lalg_zeigenderivatives" is working properly, and therefore, it is not really called anywhere in the code. It is here only for debugging purposes (perhaps it will
Definition at line 577 of file lalg_adv.F90.
| complex(real64) function, public lalg_adv_oct_m::lalg_zdni | ( | complex(real64), dimension(2) | eigenvec, |
| integer, intent(in) | alpha, | ||
| integer, intent(in) | beta | ||
| ) |
Definition at line 695 of file lalg_adv.F90.
| complex(real64) function, public lalg_adv_oct_m::lalg_zduialpha | ( | complex(real64), dimension(2) | eigenvec, |
| complex(real64), dimension(2, 2) | mmatrix, | ||
| integer, intent(in) | alpha, | ||
| integer, intent(in) | gamma, | ||
| integer, intent(in) | delta | ||
| ) |
Definition at line 701 of file lalg_adv.F90.
| complex(real64) function, public lalg_adv_oct_m::lalg_zd2ni | ( | complex(real64), dimension(2) | eigenvec, |
| complex(real64), dimension(2, 2) | mmatrix, | ||
| integer, intent(in) | alpha, | ||
| integer, intent(in) | beta, | ||
| integer, intent(in) | gamma, | ||
| integer, intent(in) | delta | ||
| ) |
Definition at line 707 of file lalg_adv.F90.
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private |
Computes the default Moore-Penrose pseudoinverse tolerance for zeroing.
We use here the value suggested here https:
Definition at line 719 of file lalg_adv.F90.
| real(real64) function, dimension(1:n, 1:n), public lalg_adv_oct_m::lalg_remove_rotation | ( | integer, intent(in) | n, |
| real(real64), dimension(1:n, 1:n), intent(in) | A | ||
| ) |
Remove rotation from affine transformation A by computing the polar decomposition and discarding the rotational part. The polar decomposition of A is given by A = U P with P = sqrt(A^T A), where U is a rotation matrix and P is a scaling matrix. This function returns P.
Definition at line 731 of file lalg_adv.F90.
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private |
Compute the Cholesky decomposition of real symmetric or complex Hermitian positive definite matrix a, dim(a) = n x n. On return a = u^T u with u upper triangular matrix.
| [in,out] | a | (n,n) |
| [in,out] | bof | Bomb on failure. |
Definition at line 816 of file lalg_adv.F90.
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private |
Computes all the eigenvalues and the eigenvectors of a real symmetric or complex Hermitian generalized definite eigenproblem, of the form \( Ax=\lambda Bx \). B is also positive definite.
For optimal performances, this uses the divide and conquer algoritm
| [in,out] | a | (n,n) |
| [in,out] | b | (n,n) |
| [out] | e | (n) |
| [in] | preserve_mat | If true, the matrix a and b on exit are the same |
| [in,out] | bof | Bomb on failure. |
Definition at line 871 of file lalg_adv.F90.
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private |
Computes all the eigenvalues and the right (left) eigenvectors of a real or complex (non-Hermitian) eigenproblem, of the form A*x=(lambda)*x.
| [in,out] | a | (n,n) |
| [out] | e | (n) |
| [in] | side | which eigenvectors ('L' or 'R') |
| [in] | sort_eigenvectors | only applies to complex version, sorts by real part |
Definition at line 982 of file lalg_adv.F90.
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private |
Computes the k lowest eigenvalues and the eigenvectors of a real symmetric or complex Hermitian generalized definite eigenproblem, of the form A*x=(lambda)*B*x. B is also positive definite.
| [in,out] | a | (n, n) |
| [in,out] | b | (n, n) |
| [out] | e | (n) |
| [out] | v | (n, n) |
| [in] | preserve_mat | If true, the matrix a and b on exit are the same |
| [in,out] | bof | Bomb on failure. |
Definition at line 1101 of file lalg_adv.F90.
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private |
Computes all eigenvalues and eigenvectors of a real symmetric or hermitian square matrix A.
| [in,out] | a | (n,n) |
| [out] | e | (n) |
| [in,out] | bof | Bomb on failure. |
Definition at line 1220 of file lalg_adv.F90.
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private |
Computes the k lowest eigenvalues and the eigenvectors of a standard symmetric-definite eigenproblem, of the form A*x=(lambda)*x. Here A is assumed to be symmetric.
| [in] | k | Number of eigenvalues requested |
| [in] | n | Dimensions of a |
| [in,out] | a | (n, n) |
| [out] | e | (n) The first k elements contain the selected eigenvalues in ascending order. |
| [out] | v | (n, k) |
| [in] | preserve_mat | If true, the matrix a and b on exit are the same |
Definition at line 1292 of file lalg_adv.F90.
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private |
Invert a real symmetric or complex Hermitian square matrix a.
| [in,out] | a | (n,n) |
Definition at line 1383 of file lalg_adv.F90.
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private |
Invert a real symmetric or complex Hermitian square matrix a.
| [in,out] | a | (n,n) |
Definition at line 1430 of file lalg_adv.F90.
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private |
Norm of a 2D matrix.
The spectral norm of a matrix \(A\) is the largest singular value of \(A\). i.e., the largest eigenvalue of the matrix \(\sqrt{\dagger{A}A}\).
| [in] | m,n | Dimensions of A |
| [in] | a | 2D matrix |
| [out] | norm2 | L2 norm of A |
Definition at line 1498 of file lalg_adv.F90.
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private |
Invert a real/complex symmetric square matrix a.
| [in,out] | a | (n,n) |
Definition at line 1532 of file lalg_adv.F90.
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private |
compute the solution to a complex system of linear equations A*X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
| [in,out] | a | (n, n) |
| [in,out] | b | (n, nrhs) |
| [out] | x | (n, nrhs) |
Definition at line 1583 of file lalg_adv.F90.
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private |
Computes the singular value decomposition of a complex MxN matrix a.
| [in,out] | a | (m,n) |
| [out] | vt | (n,n) and (m,m) |
| [out] | sg_values | (n) |
Definition at line 1661 of file lalg_adv.F90.
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private |
Computes inverse of a complex MxN matrix, a, using the SVD decomposition.
| [in,out] | a | Input (m,n) |
Definition at line 1744 of file lalg_adv.F90.
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private |
Invert a matrix with the Moore-Penrose pseudo-inverse.
SVD is used to find the U, V and Sigma matrices:
\[ A = U \Sigma V^\dagger \]
Diagonal terms in Sigma <= threshold are set to zero, and the inverse is constructed as:
\[] A^{-1} \approx V \Sigma U^\dagger \]
| [in,out] | a | Input: (m, n) |
Definition at line 1810 of file lalg_adv.F90.
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private |
Calculate the inverse of a real/complex upper triangular matrix (in unpacked storage). (lower triangular would be a trivial variant of this)
| [in,out] | a | (n,n) |
Definition at line 1874 of file lalg_adv.F90.
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private |
Definition at line 1922 of file lalg_adv.F90.
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private |
Computes all the eigenvalues and the eigenvectors of a real symmetric or complex Hermitian eigenproblem in parallel using ScaLAPACK or ELPA on all processors n: dimension of matrix a: input matrix, on exit: contains eigenvectors e: eigenvalues.
| [in,out] | a | (n,n) |
| [out] | e | (n) |
| [in,out] | bof | Bomb on failure. |
Definition at line 1997 of file lalg_adv.F90.
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private |
An interface to different method to invert a matrix.
The possible methods are: svd, dir, sym, upp For the SVD, an optional argument threshold an be specified For the direct inverse, an optional output determinant can be obtained For the symmetric matrix case, the optional argument uplo must be specified
| [in,out] | a | (n,n) |
| [out] | det | Determinant of the matrix. Direct inversion only |
| [in] | threshold | Threshold for the SVD pseudoinverse |
| [in] | uplo | Is the symmetric matrix stored in the upper or lower part? |
Definition at line 2130 of file lalg_adv.F90.
| subroutine, public lalg_adv_oct_m::zlalg_matrix_function | ( | integer, intent(in) | n, |
| complex(real64), intent(in) | factor, | ||
| complex(real64), dimension(:, :), intent(in) | a, | ||
| complex(real64), dimension(:, :), intent(inout) | fun_a, | ||
| fun, | |||
| logical, intent(in) | hermitian | ||
| ) |
This routine calculates a function of a matrix by using an eigenvalue decomposition.
For the hermitian case:
\[ A = V D V^T \implies fun(A) = V fun(D) V^T \]
and in general
\[ A = V D V^-1 \implies fun(A) = V fun(D) V^-1 \]
where \(V\) are the eigenvectors, and \(D\) is a diagonal matrix containing the eigenvalues.
In addition, this function can compute \(fun(factor*A)\) for a complex factor.
This is slow but it is simple to implement, and for the moment it does not affect performance.
| [in] | n | dimension of the matrix A |
| [in] | factor | complex factor |
| [in] | a | matrix A |
| [in,out] | fun_a | fun(A) |
| [in] | hermitian | is the matrix hermitian? |
Definition at line 2178 of file lalg_adv.F90.
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private |
Compute the Cholesky decomposition of real symmetric or complex Hermitian positive definite matrix a, dim(a) = n x n. On return a = u^T u with u upper triangular matrix.
| [in,out] | a | (n,n) |
| [in,out] | bof | Bomb on failure. |
Definition at line 2338 of file lalg_adv.F90.
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private |
Computes all the eigenvalues and the eigenvectors of a real symmetric or complex Hermitian generalized definite eigenproblem, of the form \( Ax=\lambda Bx \). B is also positive definite.
For optimal performances, this uses the divide and conquer algoritm
| [in,out] | a | (n,n) |
| [in,out] | b | (n,n) |
| [out] | e | (n) |
| [in] | preserve_mat | If true, the matrix a and b on exit are the same |
| [in,out] | bof | Bomb on failure. |
Definition at line 2393 of file lalg_adv.F90.
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private |
Computes all the eigenvalues and the right (left) eigenvectors of a real or complex (non-Hermitian) eigenproblem, of the form A*x=(lambda)*x.
| [in,out] | a | (n,n) |
| [out] | e | (n) |
| [in] | side | which eigenvectors ('L' or 'R') |
| [in] | sort_eigenvectors | only applies to complex version, sorts by real part |
Definition at line 2499 of file lalg_adv.F90.
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private |
Computes the k lowest eigenvalues and the eigenvectors of a real symmetric or complex Hermitian generalized definite eigenproblem, of the form A*x=(lambda)*B*x. B is also positive definite.
| [in,out] | a | (n, n) |
| [in,out] | b | (n, n) |
| [out] | e | (n) |
| [out] | v | (n, n) |
| [in] | preserve_mat | If true, the matrix a and b on exit are the same |
| [in,out] | bof | Bomb on failure. |
Definition at line 2618 of file lalg_adv.F90.
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private |
Computes all eigenvalues and eigenvectors of a real symmetric or hermitian square matrix A.
| [in,out] | a | (n,n) |
| [out] | e | (n) |
| [in,out] | bof | Bomb on failure. |
Definition at line 2738 of file lalg_adv.F90.
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private |
Computes the k lowest eigenvalues and the eigenvectors of a standard symmetric-definite eigenproblem, of the form A*x=(lambda)*x. Here A is assumed to be symmetric.
| [in] | k | Number of eigenvalues requested |
| [in] | n | Dimensions of a |
| [in,out] | a | (n, n) |
| [out] | e | (n) The first k elements contain the selected eigenvalues in ascending order. |
| [out] | v | (n, k) |
| [in] | preserve_mat | If true, the matrix a and b on exit are the same |
Definition at line 2810 of file lalg_adv.F90.
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private |
Invert a real symmetric or complex Hermitian square matrix a.
| [in,out] | a | (n,n) |
Definition at line 2899 of file lalg_adv.F90.
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private |
Invert a real symmetric or complex Hermitian square matrix a.
| [in,out] | a | (n,n) |
Definition at line 2946 of file lalg_adv.F90.
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private |
Norm of a 2D matrix.
The spectral norm of a matrix \(A\) is the largest singular value of \(A\). i.e., the largest eigenvalue of the matrix \(\sqrt{\dagger{A}A}\).
| [in] | m,n | Dimensions of A |
| [in] | a | 2D matrix |
| [out] | norm2 | L2 norm of A |
Definition at line 3014 of file lalg_adv.F90.
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private |
Invert a real/complex symmetric square matrix a.
| [in,out] | a | (n,n) |
Definition at line 3048 of file lalg_adv.F90.
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private |
compute the solution to a real system of linear equations A*X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
| [in,out] | a | (n, n) |
| [in,out] | b | (n, nrhs) |
| [out] | x | (n, nrhs) |
Definition at line 3099 of file lalg_adv.F90.
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private |
Computes the singular value decomposition of a real M x N matrix a.
| [in,out] | a | (m,n) |
| [out] | vt | (m,m) (n,n) |
| [out] | sg_values | (min(m,n)) |
Definition at line 3177 of file lalg_adv.F90.
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private |
Computes the inverse of a real M x N matrix, a, using the SVD decomposition.
| [in,out] | a | Input (m,n) |
Definition at line 3256 of file lalg_adv.F90.
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private |
Invert a matrix with the Moore-Penrose pseudo-inverse.
SVD is used to find the U, V and Sigma matrices:
\[ A = U \Sigma V^\dagger \]
Diagonal terms in Sigma <= threshold are set to zero, and the inverse is constructed as:
\[] A^{-1} \approx V \Sigma U^\dagger \]
| [in,out] | a | Input: (m, n) |
Definition at line 3322 of file lalg_adv.F90.
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private |
Calculate the inverse of a real/complex upper triangular matrix (in unpacked storage). (lower triangular would be a trivial variant of this)
| [in,out] | a | (n,n) |
Definition at line 3386 of file lalg_adv.F90.
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private |
Definition at line 3434 of file lalg_adv.F90.
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private |
Computes all the eigenvalues and the eigenvectors of a real symmetric or complex Hermitian eigenproblem in parallel using ScaLAPACK or ELPA on all processors n: dimension of matrix a: input matrix, on exit: contains eigenvectors e: eigenvalues.
| [in,out] | a | (n,n) |
| [out] | e | (n) |
| [in,out] | bof | Bomb on failure. |
Definition at line 3506 of file lalg_adv.F90.
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private |
An interface to different method to invert a matrix.
The possible methods are: svd, dir, sym, upp For the SVD, an optional argument threshold an be specified For the direct inverse, an optional output determinant can be obtained For the symmetric matrix case, the optional argument uplo must be specified
| [in,out] | a | (n,n) |
| [out] | det | Determinant of the matrix. Direct inversion only |
| [in] | threshold | Threshold for the SVD pseudoinverse |
| [in] | uplo | Is the symmetric matrix stored in the upper or lower part? |
Definition at line 3639 of file lalg_adv.F90.
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private |
This routine calculates a function of a matrix by using an eigenvalue decomposition.
For the hermitian case:
\[ A = V D V^T \implies fun(A) = V fun(D) V^T \]
and in general
\[ A = V D V^-1 \implies fun(A) = V fun(D) V^-1 \]
where \(V\) are the eigenvectors, and \(D\) is a diagonal matrix containing the eigenvalues.
In addition, this function can compute \(fun(factor*A)\) for a complex factor.
This is slow but it is simple to implement, and for the moment it does not affect performance.
| [in] | n | dimension of the matrix A |
| [in] | factor | complex factor |
| [in] | a | matrix A |
| [in,out] | fun_a | fun(A) |
| [in] | hermitian | is the matrix hermitian? |
Definition at line 3687 of file lalg_adv.F90.
1.9.4