Routines useful for iterative matrix calculations. More...

Data Types
interface	purify_mcweeny

Functions/Subroutines
recursive subroutine, public	determinant (matrix, det, threshold)
	Computes the determinant of a symmetric positive definite matrix using the trace of the matrix logarithm via Mercator series: det(A) = det(S)det(I+X)det(S), where S=diag(sqrt(Aii),..,sqrt(Ann)) det(I+X) = Exp(Trace(Ln(I+X))) Ln(I+X) = X - X^2/2 + X^3/3 - X^4/4 + .. The series converges only if the Frobenius norm of X is less than 1. If it is more than one we compute (recursevily) the determinant of the square root of (I+X).

subroutine, public	invert_taylor (matrix_inverse, matrix, threshold, use_inv_as_guess, norm_convergence, filter_eps, accelerator_order, max_iter_lanczos, eps_lanczos, silent)
	invert a symmetric positive definite diagonally dominant matrix

subroutine, public	invert_hotelling (matrix_inverse, matrix, threshold, use_inv_as_guess, norm_convergence, filter_eps, accelerator_order, max_iter_lanczos, eps_lanczos, silent)
	invert a symmetric positive definite matrix by Hotelling's method explicit symmetrization makes this code not suitable for other matrix types Currently a bit messy with the options, to to be cleaned soon

subroutine, public	matrix_sign_newton_schulz (matrix_sign, matrix, threshold, sign_order, iounit)
	compute the sign a matrix using Newton-Schulz iterations

subroutine, public	matrix_sign_proot (matrix_sign, matrix, threshold, sign_order)
	compute the sign a matrix using the general algorithm for the p-th root of Richters et al. Commun. Comput. Phys., 25 (2019), pp. 564-585.

subroutine, public	matrix_sign_submatrix (matrix_sign, matrix, threshold, sign_order, submatrix_sign_method)
	Submatrix method.

subroutine, public	matrix_sign_submatrix_mu_adjust (matrix_sign, matrix, mu, nelectron, threshold, variant)
	Submatrix method with internal adjustment of chemical potential.

subroutine, public	matrix_sqrt_newton_schulz (matrix_sqrt, matrix_sqrt_inv, matrix, threshold, order, eps_lanczos, max_iter_lanczos, symmetrize, converged, iounit)
	compute the sqrt of a matrix via the sign function and the corresponding Newton-Schulz iterations the order of the algorithm should be 2..5, 3 or 5 is recommended

subroutine, public	matrix_sqrt_proot (matrix_sqrt, matrix_sqrt_inv, matrix, threshold, order, eps_lanczos, max_iter_lanczos, symmetrize, converged)
	compute the sqrt of a matrix via the general algorithm for the p-th root of Richters et al. Commun. Comput. Phys., 25 (2019), pp. 564-585.

subroutine, public	matrix_exponential (matrix_exp, matrix, omega, alpha, threshold)
	...

Detailed Description

Routines useful for iterative matrix calculations.

History: 2010.10 created [Joost VandeVondele]

Author: Joost VandeVondele

Function/Subroutine Documentation

◆ determinant()

recursive subroutine, public iterate_matrix::determinant	(	type(dbcsr_type), intent(inout)	matrix,
		real(kind=dp), intent(inout)	det,
		real(kind=dp), intent(in)	threshold
	)

Computes the determinant of a symmetric positive definite matrix using the trace of the matrix logarithm via Mercator series: det(A) = det(S)det(I+X)det(S), where S=diag(sqrt(Aii),..,sqrt(Ann)) det(I+X) = Exp(Trace(Ln(I+X))) Ln(I+X) = X - X^2/2 + X^3/3 - X^4/4 + .. The series converges only if the Frobenius norm of X is less than 1. If it is more than one we compute (recursevily) the determinant of the square root of (I+X).

Parameters

matrix	...
det	- determinant
threshold	...

History: 2015.04 created [Rustam Z Khaliullin]

Author: Rustam Z. Khaliullin

Definition at line 84 of file iterate_matrix.F.

Here is the call graph for this function:

Here is the caller graph for this function:

◆ invert_taylor()

subroutine, public iterate_matrix::invert_taylor	(	type(dbcsr_type), intent(inout), target	matrix_inverse,
		type(dbcsr_type), intent(inout), target	matrix,
		real(kind=dp), intent(in)	threshold,
		logical, intent(in), optional	use_inv_as_guess,
		real(kind=dp), intent(in), optional	norm_convergence,
		real(kind=dp), intent(in), optional	filter_eps,
		integer, intent(in), optional	accelerator_order,
		integer, intent(in), optional	max_iter_lanczos,
		real(kind=dp), intent(in), optional	eps_lanczos,
		logical, intent(in), optional	silent
	)

invert a symmetric positive definite diagonally dominant matrix

Parameters

matrix_inverse	...
matrix	...
threshold	convergence threshold nased on the max abs
use_inv_as_guess	logical whether input can be used as guess for inverse
norm_convergence	convergence threshold for the 2-norm, useful for approximate solutions
filter_eps	filter_eps for matrix multiplications, if not passed nothing is filteres
accelerator_order	...
max_iter_lanczos	...
eps_lanczos	...
silent	...

History: 2010.10 created [Joost VandeVondele] 2011.10 guess option added [Rustam Z Khaliullin]

Author: Joost VandeVondele

Definition at line 284 of file iterate_matrix.F.

Here is the caller graph for this function:

◆ invert_hotelling()

subroutine, public iterate_matrix::invert_hotelling	(	type(dbcsr_type), intent(inout), target	matrix_inverse,
		type(dbcsr_type), intent(inout), target	matrix,
		real(kind=dp), intent(in)	threshold,
		logical, intent(in), optional	use_inv_as_guess,
		real(kind=dp), intent(in), optional	norm_convergence,
		real(kind=dp), intent(in), optional	filter_eps,
		integer, intent(in), optional	accelerator_order,
		integer, intent(in), optional	max_iter_lanczos,
		real(kind=dp), intent(in), optional	eps_lanczos,
		logical, intent(in), optional	silent
	)

invert a symmetric positive definite matrix by Hotelling's method explicit symmetrization makes this code not suitable for other matrix types Currently a bit messy with the options, to to be cleaned soon

Parameters

matrix_inverse	...
matrix	...
threshold	convergence threshold nased on the max abs
use_inv_as_guess	logical whether input can be used as guess for inverse
norm_convergence	convergence threshold for the 2-norm, useful for approximate solutions
filter_eps	filter_eps for matrix multiplications, if not passed nothing is filteres
accelerator_order	...
max_iter_lanczos	...
eps_lanczos	...
silent	...

History: 2010.10 created [Joost VandeVondele] 2011.10 guess option added [Rustam Z Khaliullin]

Author: Joost VandeVondele

Definition at line 470 of file iterate_matrix.F.

Here is the caller graph for this function:

◆ matrix_sign_newton_schulz()

subroutine, public iterate_matrix::matrix_sign_newton_schulz	(	type(dbcsr_type), intent(inout)	matrix_sign,
		type(dbcsr_type), intent(inout)	matrix,
		real(kind=dp), intent(in)	threshold,
		integer, intent(in), optional	sign_order,
		integer, intent(in), optional	iounit
	)

compute the sign a matrix using Newton-Schulz iterations

Parameters

matrix_sign	...
matrix	...
threshold	...
sign_order	...
iounit	...

History: 2010.10 created [Joost VandeVondele] 2019.05 extended to order byxond 2 [Robert Schade]

Author: Joost VandeVondele, Robert Schade

Definition at line 681 of file iterate_matrix.F.

Here is the caller graph for this function:

◆ matrix_sign_proot()

subroutine, public iterate_matrix::matrix_sign_proot	(	type(dbcsr_type), intent(inout)	matrix_sign,
		type(dbcsr_type), intent(inout)	matrix,
		real(kind=dp), intent(in)	threshold,
		integer, intent(in), optional	sign_order
	)

compute the sign a matrix using the general algorithm for the p-th root of Richters et al. Commun. Comput. Phys., 25 (2019), pp. 564-585.

Parameters

matrix_sign	...
matrix	...
threshold	...
sign_order	...

History: 2019.03 created [Robert Schade]

Author: Robert Schade

Definition at line 1013 of file iterate_matrix.F.

Here is the call graph for this function:

Here is the caller graph for this function:

◆ matrix_sign_submatrix()

subroutine, public iterate_matrix::matrix_sign_submatrix	(	type(dbcsr_type), intent(inout)	matrix_sign,
		type(dbcsr_type), intent(inout)	matrix,
		real(kind=dp), intent(in)	threshold,
		integer, intent(in), optional	sign_order,
		integer, intent(in)	submatrix_sign_method
	)

Submatrix method.

Parameters

matrix_sign	...
matrix	...
threshold	...
sign_order	...
submatrix_sign_method	...

History: 2019.03 created [Robert Schade] 2019.06 impl. submatrix method [Michael Lass]

Author: Robert Schade, Michael Lass

Definition at line 1364 of file iterate_matrix.F.

Here is the caller graph for this function:

◆ matrix_sign_submatrix_mu_adjust()

subroutine, public iterate_matrix::matrix_sign_submatrix_mu_adjust	(	type(dbcsr_type), intent(inout)	matrix_sign,
		type(dbcsr_type), intent(inout)	matrix,
		real(kind=dp), intent(inout)	mu,
		integer, intent(in)	nelectron,
		real(kind=dp), intent(in)	threshold,
		integer, intent(in)	variant
	)

Submatrix method with internal adjustment of chemical potential.

Parameters

matrix_sign	...
matrix	...
mu	...
nelectron	...
threshold	...
variant	...

History: 2020.05 Created [Michael Lass]

Author: Robert Schade, Michael Lass

Definition at line 1441 of file iterate_matrix.F.

Here is the caller graph for this function:

◆ matrix_sqrt_newton_schulz()

subroutine, public iterate_matrix::matrix_sqrt_newton_schulz	(	type(dbcsr_type), intent(inout)	matrix_sqrt,
		type(dbcsr_type), intent(inout)	matrix_sqrt_inv,
		type(dbcsr_type), intent(inout)	matrix,
		real(kind=dp), intent(in)	threshold,
		integer, intent(in)	order,
		real(kind=dp), intent(in)	eps_lanczos,
		integer, intent(in)	max_iter_lanczos,
		logical, optional	symmetrize,
		logical, optional	converged,
		integer, intent(in), optional	iounit
	)

compute the sqrt of a matrix via the sign function and the corresponding Newton-Schulz iterations the order of the algorithm should be 2..5, 3 or 5 is recommended

Parameters

matrix_sqrt	...
matrix_sqrt_inv	...
matrix	...
threshold	...
order	...
eps_lanczos	...
max_iter_lanczos	...
symmetrize	...
converged	...
iounit	...

History: 2010.10 created [Joost VandeVondele]

Author: Joost VandeVondele

Definition at line 1624 of file iterate_matrix.F.

Here is the caller graph for this function:

◆ matrix_sqrt_proot()

subroutine, public iterate_matrix::matrix_sqrt_proot	(	type(dbcsr_type), intent(inout)	matrix_sqrt,
		type(dbcsr_type), intent(inout)	matrix_sqrt_inv,
		type(dbcsr_type), intent(inout)	matrix,
		real(kind=dp), intent(in)	threshold,
		integer, intent(in)	order,
		real(kind=dp), intent(in)	eps_lanczos,
		integer, intent(in)	max_iter_lanczos,
		logical, optional	symmetrize,
		logical, optional	converged
	)

compute the sqrt of a matrix via the general algorithm for the p-th root of Richters et al. Commun. Comput. Phys., 25 (2019), pp. 564-585.

Parameters

matrix_sqrt	...
matrix_sqrt_inv	...
matrix	...
threshold	...
order	...
eps_lanczos	...
max_iter_lanczos	...
symmetrize	...
converged	...

History: 2019.04 created [Robert Schade]

Author: Robert Schade

Definition at line 1876 of file iterate_matrix.F.

Here is the caller graph for this function:

◆ matrix_exponential()

subroutine, public iterate_matrix::matrix_exponential	(	type(dbcsr_type), intent(inout)	matrix_exp,
		type(dbcsr_type), intent(inout)	matrix,
		real(kind=dp), intent(in)	omega,
		real(kind=dp), intent(in)	alpha,
		real(kind=dp), intent(in)	threshold
	)

...

Parameters

matrix_exp	...
matrix	...
omega	...
alpha	...
threshold	...

Definition at line 2107 of file iterate_matrix.F.

Here is the caller graph for this function:

Data Types

Functions/Subroutines

Detailed Description

Function/Subroutine Documentation

◆ determinant()

◆ invert_taylor()

◆ invert_hotelling()

◆ matrix_sign_newton_schulz()

◆ matrix_sign_proot()

◆ matrix_sign_submatrix()

◆ matrix_sign_submatrix_mu_adjust()

◆ matrix_sqrt_newton_schulz()

◆ matrix_sqrt_proot()

◆ matrix_exponential()