 |
Octopus
|
|
Octopus
|
Data Types | |
| interface | lalg_cholesky |
| interface | lalg_cholesky_pivoted |
| 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_eigensolve_tridiagonal |
| 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_rank_svd |
| interface | lalg_pseudo_inverse |
| interface | lalg_qr_factorization |
| 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) |
| 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, public | 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 | zcholesky_pivoted (A, piv, rank, tol, work, UL, bof, info) |
| Compute the Cholesky factorization with complete pivoting of a real symmetric or complex Hermitian positive-semidefinite 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 | zqr_factorization (A, jpvt, tau, info, bof) |
| Compute a QR factorization with column pivoting of matrix A: A*P = Q*R. 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 | zeigensolve_tridiagonal (n, a, e, bof, err_code) |
| Computes all eigenvalues and eigenvectors of a real symmetric tridiagonal matrix. For the Hermitian case, the matrix is assumed to be real symmetric (even if stored in complex) 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 | 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 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 | zsingular_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 | zsvd_inverse (m, n, a, threshold) |
| Computes the inverse of a real M x N matrix, a, using the SVD decomposition. More... | |
| subroutine | zlalg_pseudo_inverse (a, threshold) |
| Invert a matrix with the Moore-Penrose pseudo-inverse. More... | |
| integer function | zmatrix_rank_svd (a, preserve_mat, tol) |
| Compute the rank of the matrix A using SVD. 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, tridiagonal) |
| 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 | dcholesky_pivoted (A, piv, rank, tol, work, UL, bof, info) |
| Compute the Cholesky factorization with complete pivoting of a real symmetric or complex Hermitian positive-semidefinite 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 | dqr_factorization (A, jpvt, tau, info, bof) |
| Compute a QR factorization with column pivoting of matrix A: A*P = Q*R. 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 | deigensolve_tridiagonal (n, a, e, bof, err_code) |
| Computes all eigenvalues and eigenvectors of a real symmetric tridiagonal matrix. For the Hermitian case, the matrix is assumed to be real symmetric (even if stored in complex) 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 | 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... | |
| integer function | dmatrix_rank_svd (a, preserve_mat, tol) |
| Compute the rank of the matrix A using SVD. 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, tridiagonal) |
| This routine calculates a function of a matrix by using an eigenvalue decomposition. More... | |
|
private |
Auxiliary function.
Definition at line 258 of file lalg_adv.F90.
|
private |
| [in,out] | a | a(lda,n) |
| [out] | w | w(n) |
| [out] | vr | vl(ldvl,n), vl(ldvr,n) |
| [out] | work | work(lwork) |
| [out] | rwork | work(lwork) |
Definition at line 270 of file lalg_adv.F90.
|
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 309 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 343 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 421 of file lalg_adv.F90.
| complex(real64) function, public lalg_adv_oct_m::lalg_zdni | ( | complex(real64), dimension(2), intent(in) | eigenvec, |
| integer, intent(in) | alpha, | ||
| integer, intent(in) | beta | ||
| ) |
Definition at line 484 of file lalg_adv.F90.
| complex(real64) function, public lalg_adv_oct_m::lalg_zduialpha | ( | complex(real64), dimension(2), intent(in) | eigenvec, |
| complex(real64), dimension(2, 2), intent(in) | mmatrix, | ||
| integer, intent(in) | alpha, | ||
| integer, intent(in) | gamma, | ||
| integer, intent(in) | delta | ||
| ) |
Definition at line 490 of file lalg_adv.F90.
| complex(real64) function, public lalg_adv_oct_m::lalg_zd2ni | ( | complex(real64), dimension(2), intent(in) | eigenvec, |
| complex(real64), dimension(2, 2), intent(in) | mmatrix, | ||
| integer, intent(in) | alpha, | ||
| integer, intent(in) | beta, | ||
| integer, intent(in) | gamma, | ||
| integer, intent(in) | delta | ||
| ) |
Definition at line 496 of file lalg_adv.F90.
| pure real(real64) function, public lalg_adv_oct_m::pseudoinverse_default_tolerance | ( | integer, intent(in) | m, |
| integer, intent(in) | n, | ||
| real(real64), dimension(:), intent(in) | sg_values | ||
| ) |
Computes the default Moore-Penrose pseudoinverse tolerance for zeroing.
We use here the value suggested here https:
Definition at line 508 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 520 of file lalg_adv.F90.
|
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 605 of file lalg_adv.F90.
|
private |
Compute the Cholesky factorization with complete pivoting of a real symmetric or complex Hermitian positive-semidefinite matrix.
On return, A contains the triangular factor in the triangle selected by UL, piv contains the applied permutation, and rank contains the computed rank.
| [in] | tol | Rank tolerance; LAPACK default if absent. |
| [in,out] | work | LAPACK work array of size at least 2*n. |
| [in] | ul | Triangle to factorize; default is 'U'. |
| [in] | bof | Bomb on failure; default is true. |
| [out] | info | LAPACK status code. |
Definition at line 661 of file lalg_adv.F90.
|
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 723 of file lalg_adv.F90.
|
private |
Compute a QR factorization with column pivoting of matrix A: A*P = Q*R.
| [in,out] | a | Matrix to factorize |
| [out] | jpvt | Indices of pivots |
| [in,out] | tau | The scalar factors of the elementary reflectors |
| [out] | info | Info code returned by lapack |
| [in] | bof | Bomb on failure, by default |
Definition at line 833 of file lalg_adv.F90.
|
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 911 of file lalg_adv.F90.
|
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 1032 of file lalg_adv.F90.
|
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 1152 of file lalg_adv.F90.
|
private |
Computes all eigenvalues and eigenvectors of a real symmetric tridiagonal matrix. For the Hermitian case, the matrix is assumed to be real symmetric (even if stored in complex)
| [in,out] | a | (n,n) |
| [out] | e | (n) |
| [in,out] | bof | Bomb on failure. |
Definition at line 1225 of file lalg_adv.F90.
|
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 1304 of file lalg_adv.F90.
|
private |
Invert a real symmetric or complex Hermitian square matrix a.
| [in,out] | a | (n,n) |
Definition at line 1396 of file lalg_adv.F90.
|
private |
Invert a real symmetric or complex Hermitian square matrix a.
| [in,out] | a | (n,n) |
Definition at line 1443 of file lalg_adv.F90.
|
private |
Invert a real/complex symmetric square matrix a.
| [in,out] | a | (n,n) |
Definition at line 1502 of file lalg_adv.F90.
|
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 1550 of file lalg_adv.F90.
|
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 1650 of file lalg_adv.F90.
|
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 1748 of file lalg_adv.F90.
|
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 1811 of file lalg_adv.F90.
|
private |
Compute the rank of the matrix A using SVD.
The rank is equal to the number of non-zero singular values in the diagonal matrix of the SVD decomposition.
| [in,out] | a | Input: (m, n) |
| [in] | preserve_mat | Preserve input A |
| [in] | tol | Tolerance defining |
Definition at line 1877 of file lalg_adv.F90.
|
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 1918 of file lalg_adv.F90.
|
private |
Definition at line 1966 of file lalg_adv.F90.
|
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 2043 of file lalg_adv.F90.
|
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 2066 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, | ||
| logical, intent(in), optional | tridiagonal | ||
| ) |
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? |
| [in] | tridiagonal | is the matrix tridiagonal? |
Definition at line 2114 of file lalg_adv.F90.
|
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 2296 of file lalg_adv.F90.
|
private |
Compute the Cholesky factorization with complete pivoting of a real symmetric or complex Hermitian positive-semidefinite matrix.
On return, A contains the triangular factor in the triangle selected by UL, piv contains the applied permutation, and rank contains the computed rank.
| [in] | tol | Rank tolerance; LAPACK default if absent. |
| [in,out] | work | LAPACK work array of size at least 2*n. |
| [in] | ul | Triangle to factorize; default is 'U'. |
| [in] | bof | Bomb on failure; default is true. |
| [out] | info | LAPACK status code. |
Definition at line 2352 of file lalg_adv.F90.
|
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 2414 of file lalg_adv.F90.
|
private |
Compute a QR factorization with column pivoting of matrix A: A*P = Q*R.
| [in,out] | a | Matrix to factorize |
| [out] | jpvt | Indices of pivots |
| [in,out] | tau | The scalar factors of the elementary reflectors |
| [out] | info | Info code returned by lapack |
| [in] | bof | Bomb on failure, by default |
Definition at line 2520 of file lalg_adv.F90.
|
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 2598 of file lalg_adv.F90.
|
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 2719 of file lalg_adv.F90.
|
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 2840 of file lalg_adv.F90.
|
private |
Computes all eigenvalues and eigenvectors of a real symmetric tridiagonal matrix. For the Hermitian case, the matrix is assumed to be real symmetric (even if stored in complex)
| [in,out] | a | (n,n) |
| [out] | e | (n) |
| [in,out] | bof | Bomb on failure. |
Definition at line 2911 of file lalg_adv.F90.
|
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 2990 of file lalg_adv.F90.
|
private |
Invert a real symmetric or complex Hermitian square matrix a.
| [in,out] | a | (n,n) |
Definition at line 3080 of file lalg_adv.F90.
|
private |
Invert a real symmetric or complex Hermitian square matrix a.
| [in,out] | a | (n,n) |
Definition at line 3127 of file lalg_adv.F90.
|
private |
Invert a real/complex symmetric square matrix a.
| [in,out] | a | (n,n) |
Definition at line 3186 of file lalg_adv.F90.
|
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 3234 of file lalg_adv.F90.
|
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 3334 of file lalg_adv.F90.
|
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 3424 of file lalg_adv.F90.
|
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 3487 of file lalg_adv.F90.
|
private |
Compute the rank of the matrix A using SVD.
The rank is equal to the number of non-zero singular values in the diagonal matrix of the SVD decomposition.
| [in,out] | a | Input: (m, n) |
| [in] | preserve_mat | Preserve input A |
| [in] | tol | Tolerance defining |
Definition at line 3553 of file lalg_adv.F90.
|
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 3594 of file lalg_adv.F90.
|
private |
Definition at line 3642 of file lalg_adv.F90.
|
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 3716 of file lalg_adv.F90.
|
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 3739 of file lalg_adv.F90.
|
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? |
| [in] | tridiagonal | is the matrix tridiagonal? |
Definition at line 3787 of file lalg_adv.F90.
1.9.4