linalg_cholesky

public interface cholesky_factor

private pure function cholesky_factor_dbl(a, upper) result(rst)

Computes the Cholesky factorization of a symmetric, positive definite matrix.

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in),		dimension(:,:)	::	a	The N-by-N matrix to factor.
logical,	intent(in),	optional		::	upper	An optional input that, if specified, provides control over whether the factorization is computed as $A = U^T U$ (set to true), or as $A = L L^T$ (set to false). The default is true such that $A = U^T U$ .

Return Value real(kind=real64), allocatable, dimension(:,:)

The factored matrix.

private pure function cholesky_factor_cmplx(a, upper) result(rst)

Computes the Cholesky factorization of a symmetric, positive definite matrix.

Arguments

Type	Intent	Optional	Attributes		Name
complex(kind=real64),	intent(in),		dimension(:,:)	::	a	The N-by-N matrix to factor.
logical,	intent(in),	optional		::	upper	An optional input that, if specified, provides control over whether the factorization is computed as $A = U^H U$ (set to true), or as $A = L L^H$ (set to false). The default is true such that $A = U^H U$ .

Return Value complex(kind=real64), allocatable, dimension(:,:)

The factored matrix.

public interface cholesky_rank1_downdate

private pure subroutine cholesky_rank1_downdate_dbl(r, u)

Computes the rank 1 downdate to a Cholesky factored matrix $A = R^T R$ such that $A_1 = A - \vec{u} \vec{u}^T$ . This operation only works if the new matrix $A_1$ is positive definite.

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(inout),		dimension(:,:)	::	r	On input, the N-by-N upper triangular matrix $R$ . On output, the updated matrix $R_1$ .
real(kind=real64),	intent(inout),		dimension(:)	::	u	On input, the N-element vector $\vec{u}$ . On output, the rotation sines used to transform $R$ to $R_1$ .

private pure subroutine cholesky_rank1_downdate_cmplx(r, u)

Computes the rank 1 downdate to a Cholesky factored matrix $A = R^H R$ such that $A_1 = A - \vec{u} \vec{u}^H$ . This operation only works if the new matrix $A_1$ is positive definite.

Arguments

Type	Intent	Optional	Attributes		Name
complex(kind=real64),	intent(inout),		dimension(:,:)	::	r	On input, the N-by-N upper triangular matrix $R$ . On output, the updated matrix $R_1$ .
complex(kind=real64),	intent(inout),		dimension(:)	::	u	On input, the N-element vector $\vec{u}$ . On output, the rotation sines used to transform $R$ to $R_1$ .

public interface cholesky_rank1_update

private pure subroutine cholesky_rank1_update_dbl(r, u)

Computes the rank 1 update to a Cholesky factored matrix $A = R^T R$ such that $A_1 = A + \vec{u} \vec{u}^T$ .

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(inout),		dimension(:,:)	::	r	On input, the N-by-N upper triangular matrix $R$ . On output, the updated matrix $R_1$ .
real(kind=real64),	intent(inout),		dimension(:)	::	u	On input, the N-element vector $\vec{u}$ . On output, the rotation sines used to transform $R$ to $R_1$ .

private pure subroutine cholesky_rank1_update_cmplx(r, u)

Computes the rank 1 update to a Cholesky factored matrix $A = R^H R$ such that $A_1 = A + \vec{u} \vec{u}^H$ .

Arguments

Type	Intent	Optional	Attributes		Name
complex(kind=real64),	intent(inout),		dimension(:,:)	::	r	On input, the N-by-N upper triangular matrix $R$ . On output, the updated matrix $R_1$ .
complex(kind=real64),	intent(inout),		dimension(:)	::	u	On input, the N-element vector $\vec{u}$ . On output, the rotation sines used to transform $R$ to $R_1$ .

public interface solve_cholesky

private pure function solve_cholesky_mtx(upper, a, b) result(x)

Solves the system of Cholesky factored equations $A X = R^T R X = B$ or $A X = L L^T X = B$ .

Arguments

Type	Intent	Attributes		Name
logical,	intent(in)		::	upper	Set to true if $A$ is factored such that $A = R^T R$ ; else, set to false if $A$ is factored such that $A = L L^T$ .
real(kind=real64),	intent(in),	dimension(:,:)	::	a	The N-by-N Cholesky factored matrix as returned by cholesky_factor.
real(kind=real64),	intent(in),	dimension(:,:)	::	b	The N-by-NRHS matrix $B$ .

Return Value real(kind=real64), allocatable, dimension(:,:)

The resulting N-by-NRHS matrix $X$ .

private pure function solve_cholesky_mtx_cmplx(upper, a, b) result(x)

Solves the system of Cholesky factored equations $A X = R^H R X = B$ or $A X = L L^H X = B$ .

Arguments

Type	Intent	Attributes		Name
logical,	intent(in)		::	upper	Set to true if $A$ is factored such that $A = R^H R$ ; else, set to false if $A$ is factored such that $A = L L^H$ .
complex(kind=real64),	intent(in),	dimension(:,:)	::	a	The N-by-N Cholesky factored matrix as returned by cholesky_factor.
complex(kind=real64),	intent(in),	dimension(:,:)	::	b	The N-by-NRHS matrix $B$ .

Return Value complex(kind=real64), allocatable, dimension(:,:)

The resulting N-by-NRHS matrix $X$ .

private pure function solve_cholesky_vec(upper, a, b) result(x)

Solves the system of Cholesky factored equations $A \vec{x} = R^T R \vec{x} = \vec{b}$ or $A \vec{x} = L L^T \vec{x} = \vec{b}$ .

Arguments

Type	Intent	Attributes		Name
logical,	intent(in)		::	upper	Set to true if $A$ is factored such that $A = R^T R$ ; else, set to false if $A$ is factored such that $A = L L^T$ .
real(kind=real64),	intent(in),	dimension(:,:)	::	a	The N-by-N Cholesky factored matrix as returned by cholesky_factor.
real(kind=real64),	intent(in),	dimension(:)	::	b	The N-element vector $\vec{b}$ .

Return Value real(kind=real64), allocatable, dimension(:)

The resulting N-element vector $\vec{x}$ .

private pure function solve_cholesky_vec_cmplx(upper, a, b) result(x)

Solves the system of Cholesky factored equations $A \vec{x} = R^H R \vec{x} = \vec{b}$ or $A \vec{x} = L L^H \vec{x} = \vec{b}$ .

Arguments

Type	Intent	Attributes		Name
logical,	intent(in)		::	upper	Set to true if $A$ is factored such that $A = R^H R$ ; else, set to false if $A$ is factored such that $A = L L^H$ .
complex(kind=real64),	intent(in),	dimension(:,:)	::	a	The N-by-N Cholesky factored matrix as returned by cholesky_factor.
complex(kind=real64),	intent(in),	dimension(:)	::	b	The N-element vector $\vec{b}$ .

Return Value complex(kind=real64), allocatable, dimension(:)

The resulting N-element vector $\vec{x}$ .

linalg_cholesky Module

Contents

Interfaces

Uses

Interfaces

public interface cholesky_factor

private pure function cholesky_factor_dbl(a, upper) result(rst)

Arguments

Return Value real(kind=real64), allocatable, dimension(:,:)

private pure function cholesky_factor_cmplx(a, upper) result(rst)

Arguments

Return Value complex(kind=real64), allocatable, dimension(:,:)

public interface cholesky_rank1_downdate

private pure subroutine cholesky_rank1_downdate_dbl(r, u)

Arguments

private pure subroutine cholesky_rank1_downdate_cmplx(r, u)

Arguments

public interface cholesky_rank1_update

private pure subroutine cholesky_rank1_update_dbl(r, u)

Arguments

private pure subroutine cholesky_rank1_update_cmplx(r, u)

Arguments

public interface solve_cholesky

private pure function solve_cholesky_mtx(upper, a, b) result(x)

Arguments

Return Value real(kind=real64), allocatable, dimension(:,:)

private pure function solve_cholesky_mtx_cmplx(upper, a, b) result(x)

Arguments

Return Value complex(kind=real64), allocatable, dimension(:,:)

private pure function solve_cholesky_vec(upper, a, b) result(x)

Arguments

Return Value real(kind=real64), allocatable, dimension(:)

private pure function solve_cholesky_vec_cmplx(upper, a, b) result(x)

Arguments

Return Value complex(kind=real64), allocatable, dimension(:)