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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_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, public | lalg_zpseudoinverse (n, mat, imat) |
| 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 | 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 NxN matrix a(:,:) More... | |
| subroutine | dsvd_inverse (m, n, a, threshold) |
| computes inverse of a real NxN matrix a(:,:) using the SVD decomposition 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 236 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 248 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 275 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 309 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 387 of file lalg_adv.F90.
| subroutine, public lalg_adv_oct_m::lalg_zeigenderivatives | ( | integer, intent(in) | n, |
| complex(real64), dimension(:, :), intent(in) | mat, | ||
| complex(real64), dimension(:, :), intent(out) | zeigenvec, | ||
| complex(real64), dimension(:), intent(out) | zeigenval, | ||
| complex(real64), dimension(:, :, :), intent(out) | 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 462 of file lalg_adv.F90.
| subroutine, public lalg_adv_oct_m::lalg_zpseudoinverse | ( | integer, intent(in) | n, |
| complex(real64), dimension(:, :), intent(in) | mat, | ||
| complex(real64), dimension(:, :), intent(out) | imat | ||
| ) |
Computes the Moore-Penrose pseudoinverse of a
Definition at line 514 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 573 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 691 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 697 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 703 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 715 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 727 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 812 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.
| [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 865 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 967 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 1086 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 1205 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 1277 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 1368 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 1415 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 1483 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 1517 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 1568 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 1647 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 | (m,n); a will be replaced by its inverse transposed |
Definition at line 1729 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 1784 of file lalg_adv.F90.
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private |
Definition at line 1832 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 1907 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 2040 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 2088 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 2248 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.
| [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 2301 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 2403 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 2522 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 2642 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 2714 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 2803 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 2850 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 2918 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 2952 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 3003 of file lalg_adv.F90.
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private |
computes the singular value decomposition of a real NxN matrix a(:,:)
| [in,out] | a | (m,n) |
| [out] | vt | (m,m) (n,n) |
| [out] | sg_values | (min(m,n)) |
Definition at line 3082 of file lalg_adv.F90.
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private |
computes inverse of a real NxN matrix a(:,:) using the SVD decomposition
| [in,out] | a | (m,n); a will be replaced by its inverse |
Definition at line 3160 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 3214 of file lalg_adv.F90.
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private |
Definition at line 3262 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 3334 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 3467 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 3515 of file lalg_adv.F90.
1.9.4