Algebraic operations with hierarchical matrices. More...

Data Structures
struct	_haccum
	Accumulator for H-matrix products. More...

Typedefs
typedef struct _haccum	haccum
	Accumulator for H-matrix products.

typedef haccum *	phaccum
	Pointer to haccum object.

typedef const haccum *	pchaccum
	Pointer to constant haccum object.

typedef struct _hprodentry	hprodentry
	Unhandled H-matrix products.

typedef hprodentry *	phprodentry
	Points to hprodentry object.

Functions
void	trunc_rkmatrix (pctruncmode tm, real eps, prkmatrix r)
	Truncate an rkmatrix, $A \gets \operatorname{trunc}(A,\epsilon)$ . More...

void	add_amatrix_rkmatrix (field alpha, bool atrans, pcamatrix a, pctruncmode tm, real eps, prkmatrix b)
	Truncated addition of a matrix to a low-rank matrices in rkmatrix representation, $B \gets \operatorname{trunc}(B+\alpha A,\epsilon)$ . More...

void	add_rkmatrix_amatrix (field alpha, bool atrans, pcrkmatrix a, pamatrix b)
	Truncated addition of a full matrix to a low-rank matrix. More...

void	add_hmatrix_amatrix (field alpha, bool atrans, pchmatrix a, pamatrix b)
	Addition of a hierarchical matrix to a matrix, $B \gets B+\alpha A$ . More...

void	add_rkmatrix (field alpha, pcrkmatrix src, pctruncmode tm, real eps, prkmatrix trg)
	Truncated addition of two low-rank matrices in rkmatrix representation, $A \gets \operatorname{trunc}(A+\alpha B,\epsilon)$ . More...

void	add_amatrix_hmatrix (field alpha, bool atrans, pcamatrix a, pctruncmode tm, real eps, phmatrix b)
	Truncated addition of two low-rank matrices in rkmatrix. More...

void	add_lower_amatrix_hmatrix (field alpha, bool atrans, pcamatrix a, pctruncmode tm, real eps, phmatrix b)
	Locally truncated addition of a matrix in amatrix representation to the lower triangular part of a hierarchical matrix in hmatrix representation, $\operatorname{lower}(B) \gets \operatorname{blocktrunc}(\operatorname{lower}(B + \alpha A),\epsilon)$ . More...

void	add_rkmatrix_hmatrix (field alpha, pcrkmatrix a, pctruncmode tm, real eps, phmatrix b)
	Locally truncated addition of a low-rank matrix in rkmatrix representation to a hierarchical matrix in hmatrix representation, $B \gets \operatorname{blocktrunc}(B + \alpha A,\epsilon)$ . More...

void	add_lower_rkmatrix_hmatrix (field alpha, pcrkmatrix a, pctruncmode tm, real eps, phmatrix b)
	Locally truncated addition of a low-rank matrix in rkmatrix representation to the lower triangular part of a hierarchical matrix in hmatrix representation, $\operatorname{lower}(B) \gets \operatorname{blocktrunc}(\operatorname{lower}(B + \alpha A),\epsilon)$ . More...

void	add_hmatrix (field alpha, pchmatrix a, pctruncmode tm, real eps, phmatrix b)
	Locally truncated addition of a hierarchical matrix to a hierarchical matrix representation, $B \gets \operatorname{blocktrunc}(B + \alpha A,\epsilon)$ . More...

phmatrix	split_sub_amatrix (pcamatrix f, pccluster rc, pccluster cc, bool rsplit, bool csplit)
	Split an amatrix into submatrices. More...

phmatrix	split_rkmatrix (pcrkmatrix r, pccluster rc, pccluster cc, bool rsplit, bool csplit, bool copy)
	Split an rkmatrix into submatrices. More...

phmatrix	split_sub_rkmatrix (pcrkmatrix r, pccluster rc, pccluster cc, bool rsplit, bool csplit)
	Split an rkmatrix into submatrices referencing the original matrix. More...

void	merge_rkmatrix (bool colmerge, pcrkmatrix src, pctruncmode tm, real eps, prkmatrix trg)
	Split a leaf hmatrix into submatrices. More...

prkmatrix	merge_hmatrix_rkmatrix (pchmatrix s, pctruncmode tm, real eps)
	Merge a depth-one hmatrix containing only rkmatrix submatrices into a new rkmatrix, $A \gets \operatorname{succtrunc}\left(\begin{pmatrix} B_{11} & \ldots & B_{1m}\\ \vdots & \ddots & \vdots\\ B_{n1} & \ldots & B_{nm} \end{pmatrix},\epsilon\right)$ . More...

void	addmul_rkmatrix_amatrix_amatrix (field alpha, bool atrans, pcrkmatrix a, bool btrans, pcamatrix b, bool ctrans, pamatrix c)
	Multiply a matrix by an rkmatrix, $C \gets C + \alpha A B$ . More...

void	addmul_hmatrix_amatrix_amatrix (field alpha, bool atrans, pchmatrix a, bool btrans, pcamatrix bp, bool ctrans, pamatrix cp)
	Multiply a matrix by a hmatrix, $C \gets C + \alpha A B$ . More...

void	addmul_hmatrix (field alpha, bool xtrans, pchmatrix x, bool ytrans, pchmatrix y, pctruncmode tm, real eps, phmatrix z)
	Multiply two H-matrices, $Z \gets \operatorname{succtrunc}(Z + \alpha X Y,\epsilon)$ . More...

void	addmul_lower_hmatrix (field alpha, bool xtrans, pchmatrix x, bool ytrans, pchmatrix y, pctruncmode tm, real eps, phmatrix z)
	Multiply two H-matrices, computing only the lower triangular part of the result, $\operatorname{lower}(Z) \gets \operatorname{succtrunc}(\operatorname{lower}(Z + \alpha X Y),\epsilon)$ . More...

void	invert_hmatrix (phmatrix a, phmatrix work, pctruncmode tm, real eps)
	Invert an H-matrix, $A \gets \operatorname{succtrunc}(A^{-1},\epsilon)$ . More...

phmatrix	clone_lower_hmatrix (bool aunit, pchmatrix a)
	Create a duplicate of the lower triangular part of an H-matrix. More...

phmatrix	clone_upper_hmatrix (bool aunit, pchmatrix a)
	Create a duplicate of the upper triangular part of an H-matrix. More...

void	triangularinvmul_hmatrix_avector (bool alower, bool aunit, bool atrans, pchmatrix a, pavector xp)
	Solve a triangular system, $x \gets R^{-1} x$ or $x \gets R^{-*} x$ . More...

void	triangularsolve_hmatrix_avector (bool alower, bool aunit, bool atrans, pchmatrix a, pavector x)
	Solve a triangular system $x \gets A^{-1} x$ or $x \gets (A^*)^{-1} x$ . More...

void	triangularinvmul_hmatrix_amatrix (bool alower, bool aunit, bool atrans, pchmatrix a, bool xtrans, pamatrix xp)
	Solve a triangular system $X \gets A^{-1} X$ or $X \gets (A^*)^{-1} X$ . More...

void	triangularinvmul_hmatrix (bool alower, bool aunit, bool atrans, pchmatrix a, pctruncmode tm, real eps, bool xtrans, phmatrix xp)
	Solve a triangular system, $X \gets \operatorname{succtrunc}(A^{-1} X, \epsilon)$ or $X \gets \operatorname{succtrunc}(A^{-*} X, \epsilon)$ . More...

void	triangularinvmul_amatrix_hmatrix (bool alower, bool aunit, bool atrans, pcamatrix a, pctruncmode tm, real eps, bool xtrans, phmatrix xp)
	Solve a triangular system, $X \gets \operatorname{succtrunc}(A^{-1} X, \epsilon)$ or $X \gets \operatorname{succtrunc}(A^{-*} X, \epsilon)$ . More...

void	triangularmul_hmatrix_avector (bool alower, bool aunit, bool atrans, pchmatrix a, pavector xp)
	Evaluate a triangular system $x \gets A x$ or $x \gets A^* x$ . More...

void	triangulareval_hmatrix_avector (bool alower, bool aunit, bool atrans, pchmatrix a, pavector x)
	Evaluate a triangular system $x \gets A x$ or $x \gets A^* x$ . More...

void	triangularmul_hmatrix_amatrix (bool alower, bool aunit, bool atrans, pchmatrix a, bool xtrans, pamatrix xp)
	Evaluate a triangular system $X \gets A X$ or $X \gets A^* X$ . More...

void	triangularmul_hmatrix (bool alower, bool aunit, bool atrans, pchmatrix a, pctruncmode tm, real eps, bool xtrans, phmatrix xp)
	Evaluate a triangular system, $X \gets \operatorname{succtrunc}(A X, \epsilon)$ or $X \gets \operatorname{succtrunc}(A^{*} X, \epsilon)$ . More...

void	lrdecomp_hmatrix (phmatrix a, pctruncmode tm, real eps)
	Compute the LR factorization, $A \approx L R$ . More...

void	lrsolve_n_hmatrix_avector (pchmatrix a, pavector x)
	Solve the linear systems using the LR factorization provided by lrdecomp_hmatrix. More...

void	lrsolve_t_hmatrix_avector (pchmatrix a, pavector x)
	Solve the linear systems using the LR factorization provided by lrdecomp_hmatrix. More...

void	lrsolve_hmatrix_avector (bool atrans, pchmatrix a, pavector x)
	Solve the linear systems or using the LR factorization provided by lrdecomp_hmatrix. More...

void	lreval_n_hmatrix_avector (pchmatrix a, pavector x)
	Evaluate the linear system $x \gets A x$ using the LR factorization provided by lrdecomp_hmatrix. More...

void	lreval_t_hmatrix_avector (pchmatrix a, pavector x)
	Evaluate the linear system $x \gets A^* x$ using the LR factorization provided by lrdecomp_hmatrix. More...

void	lreval_hmatrix_avector (bool atrans, pchmatrix a, pavector x)
	Evaluate the linear system $x \gets A x$ or $x \gets A^* x$ using the LR factorization provided by lrdecomp_hmatrix. More...

void	choldecomp_hmatrix (phmatrix a, pctruncmode tm, real eps)
	Compute the Cholesky factorization, $A \approx L L^*$ . More...

void	cholsolve_hmatrix_avector (pchmatrix a, pavector x)
	Solve the linear system using the Cholesky factorization provided by choldecomp_hmatrix. More...

void	choleval_hmatrix_avector (pchmatrix a, pavector x)
	Evaluate the linear system $x \gets A x$ using the Cholesky factorization provided by choldecomp_hmatrix. More...

phaccum	new_haccum (phmatrix z, pctruncmode tm, real eps)
	Create a new H-matrix accumulator. More...

void	del_haccum (phaccum ha)
	Delete an accumulator. More...

void	addproduct_haccum (field alpha, bool xtrans, pchmatrix x, bool ytrans, pchmatrix y, phaccum za)
	Add a product $\alpha X Y$ to an accumulator. More...

phaccum *	split_haccum (phmatrix z, phaccum za)
	Create H-matrix accumulators for submatrices. More...

void	flush_haccum (phaccum ha)
	Flush an H-matrix accumulator, i.e., add all accumulated products to the target matrix and clear the accumulator. More...

void	add_amatrix_destructive_rkmatrix (field alpha, bool atrans, pamatrix a, pctruncmode tm, real eps, prkmatrix b)
	Truncated addition of a matrix to a low-rank matrices in rkmatrix representation, $B \gets \operatorname{trunc}(B+\alpha A,\epsilon)$ . More...

void	add_amatrix_destructive_hmatrix (field alpha, bool atrans, pamatrix a, pctruncmode tm, real eps, phmatrix b)
	Locally truncated addition of a matrix in amatrix representation to a hierarchical matrix in hmatrix representation, $B \gets \operatorname{blocktrunc}(B + \alpha A,\epsilon)$ . More...

void	addmul2_hmatrix (field alpha, bool xtrans, pchmatrix x, bool ytrans, pchmatrix y, pctruncmode tm, real eps, phmatrix z)
	Multiply two H-matrices using accumulators, $Z \gets \operatorname{succtrunc}(Z + \alpha X Y,\epsilon)$ . More...

void	lowersolve_haccum (bool aunit, bool atrans, pchmatrix a, bool xtrans, phaccum xa)
	Solve a lower triangular system using accumulators, $X \gets \operatorname{succtrunc}(L^{-1} X, \epsilon)$ or $X \gets \operatorname{succtrunc}(L^{-*} X, \epsilon)$ . More...

void	lowersolve2_hmatrix_hmatrix (bool aunit, bool atrans, pchmatrix a, pctruncmode tm, real eps, bool xtrans, phmatrix x)
	Solve a lower triangular system using accumulators, $X \gets \operatorname{succtrunc}(L^{-1} X, \epsilon)$ or $X \gets \operatorname{succtrunc}(L^{-*} X, \epsilon)$ . More...

void	uppersolve_haccum (bool aunit, bool atrans, pchmatrix a, bool xtrans, phaccum xa)
	Solve an upper triangular system using accumulators, $X \gets \operatorname{succtrunc}(R^{-1} X, \epsilon)$ or $X \gets \operatorname{succtrunc}(R^{-*} X, \epsilon)$ . More...

void	uppersolve2_hmatrix_hmatrix (bool aunit, bool atrans, pchmatrix a, pctruncmode tm, real eps, bool xtrans, phmatrix x)
	Solve an upper triangular system using accumulators, $X \gets \operatorname{succtrunc}(R^{-1} X, \epsilon)$ or $X \gets \operatorname{succtrunc}(R^{-*} X, \epsilon)$ . More...

void	lrdecomp_haccum (phaccum aa)
	Compute the LR factorization using accumulators, $A \approx L R$ . More...

void	lrdecomp2_hmatrix (phmatrix a, pctruncmode tm, real eps)
	Compute the LR factorization using accumulators, $A \approx L R$ . More...

void	choldecomp_haccum (phaccum aa)
	Compute the Cholesky factorization using accumulators, $A \approx L L^*$ . More...

void	choldecomp2_hmatrix (phmatrix a, pctruncmode tm, real eps)
	Compute the Cholesky factorization using accumulators, $A \approx L L^*$ . More...

Detailed Description

Algebraic operations with hierarchical matrices.

Function Documentation

void add_amatrix_destructive_hmatrix	(	field	alpha,
		bool	atrans,
		pamatrix	a,
		pctruncmode	tm,
		real	eps,
		phmatrix	b
	)

Locally truncated addition of a matrix in amatrix representation to a hierarchical matrix in hmatrix representation, $B \gets \operatorname{blocktrunc}(B + \alpha A,\epsilon)$ .

Attention: The matrix is overwritten by intermediate results.

Parameters

alpha	Scaling factor $\alpha$ .
atrans	Set if is to be used instead of .
a	Matrix .
tm	Truncation mode.
eps	Truncation accuracy $\epsilon$ .
b	Target hmatrix .

void add_amatrix_destructive_rkmatrix	(	field	alpha,
		bool	atrans,
		pamatrix	a,
		pctruncmode	tm,
		real	eps,
		prkmatrix	b
	)

Truncated addition of a matrix to a low-rank matrices in rkmatrix representation, $B \gets \operatorname{trunc}(B+\alpha A,\epsilon)$ .

Attention: The matrix is overwritten by intermediate results.

Parameters

alpha	Scaling factor $\alpha$ .
atrans	Set if is to be used instead of .
a	Matrix .
tm	Truncation mode.
eps	Truncation accuracy $\epsilon$ .
b	Target matrix .

void add_amatrix_hmatrix	(	field	alpha,
		bool	atrans,
		pcamatrix	a,
		pctruncmode	tm,
		real	eps,
		phmatrix	b
	)

Truncated addition of two low-rank matrices in rkmatrix.

*Locally truncated addition of a matrix in amatrix representation to a hierarchical matrix in hmatrix representation, $B \gets \operatorname{blocktrunc}(B + \alpha A,\epsilon)$ .

Parameters

alpha	Scaling factor $\alpha$ .
atrans	Set if is to be used instead of .
a	Matrix .
tm	Truncation mode.
eps	Truncation accuracy $\epsilon$ .
b	Target hmatrix .

void add_amatrix_rkmatrix	(	field	alpha,
		bool	atrans,
		pcamatrix	a,
		pctruncmode	tm,
		real	eps,
		prkmatrix	b
	)

Truncated addition of a matrix to a low-rank matrices in rkmatrix representation, $B \gets \operatorname{trunc}(B+\alpha A,\epsilon)$ .

Parameters

alpha	Scaling factor $\alpha$ .
atrans	Set if is to be used instead of .
a	Matrix .
tm	Truncation mode.
eps	Truncation accuracy $\epsilon$ .
b	Target matrix .

void add_hmatrix	(	field	alpha,
		pchmatrix	a,
		pctruncmode	tm,
		real	eps,
		phmatrix	b
	)

Locally truncated addition of a hierarchical matrix to a hierarchical matrix representation, $B \gets \operatorname{blocktrunc}(B + \alpha A,\epsilon)$ .

Parameters

alpha	Scaling factor $\alpha$ .
a	hmatrix .
tm	Truncation mode.
eps	Truncation accuracy $\epsilon$ .
b	Target hmatrix .

void add_hmatrix_amatrix	(	field	alpha,
		bool	atrans,
		pchmatrix	a,
		pamatrix	b
	)

Addition of a hierarchical matrix to a matrix, $B \gets B+\alpha A$ .

Parameters

alpha	Scaling factor $\alpha$ .
atrans	Set if is to be used instead of .
a	Matrix .
b	Target matrix .

void add_lower_amatrix_hmatrix	(	field	alpha,
		bool	atrans,
		pcamatrix	a,
		pctruncmode	tm,
		real	eps,
		phmatrix	b
	)

Locally truncated addition of a matrix in amatrix representation to the lower triangular part of a hierarchical matrix in hmatrix representation, $\operatorname{lower}(B) \gets \operatorname{blocktrunc}(\operatorname{lower}(B + \alpha A),\epsilon)$ .

Parameters

alpha	Scaling factor $\alpha$ .
atrans	Set if is to be used instead of .
a	Matrix .
tm	Truncation mode.
eps	Truncation accuracy $\epsilon$ .
b	Target hmatrix .

void add_lower_rkmatrix_hmatrix	(	field	alpha,
		pcrkmatrix	a,
		pctruncmode	tm,
		real	eps,
		phmatrix	b
	)

Locally truncated addition of a low-rank matrix in rkmatrix representation to the lower triangular part of a hierarchical matrix in hmatrix representation, $\operatorname{lower}(B) \gets \operatorname{blocktrunc}(\operatorname{lower}(B + \alpha A),\epsilon)$ .

Parameters

alpha	Scaling factor $\alpha$ .
a	Low-rank matrix .
tm	Truncation mode.
eps	Truncation accuracy $\epsilon$ .
b	Target hmatrix .

void add_rkmatrix	(	field	alpha,
		pcrkmatrix	src,
		pctruncmode	tm,
		real	eps,
		prkmatrix	trg
	)

Truncated addition of two low-rank matrices in rkmatrix representation, $A \gets \operatorname{trunc}(A+\alpha B,\epsilon)$ .

Parameters

alpha	Scaling factor $\alpha$ .
src	Source matrix .
tm	Truncation mode.
eps	Truncation accuracy $\epsilon$ .
trg	Target matrix .

void add_rkmatrix_amatrix	(	field	alpha,
		bool	atrans,
		pcrkmatrix	a,
		pamatrix	b
	)

Truncated addition of a full matrix to a low-rank matrix.

*Addition of a low-rank matrix to a matrix, $B \gets B+\alpha A$ .

Parameters

alpha	Scaling factor $\alpha$ .
atrans	Set if is to be used instead of .
a	Matrix .
b	Target matrix .

void add_rkmatrix_hmatrix	(	field	alpha,
		pcrkmatrix	a,
		pctruncmode	tm,
		real	eps,
		phmatrix	b
	)

Locally truncated addition of a low-rank matrix in rkmatrix representation to a hierarchical matrix in hmatrix representation, $B \gets \operatorname{blocktrunc}(B + \alpha A,\epsilon)$ .

Parameters

alpha	Scaling factor $\alpha$ .
a	Low-rank matrix .
tm	Truncation mode.
eps	Truncation accuracy $\epsilon$ .
b	Target hmatrix .

void addmul2_hmatrix	(	field	alpha,
		bool	xtrans,
		pchmatrix	x,
		bool	ytrans,
		pchmatrix	y,
		pctruncmode	tm,
		real	eps,
		phmatrix	z
	)

Multiply two H-matrices using accumulators, $Z \gets \operatorname{succtrunc}(Z + \alpha X Y,\epsilon)$ .

Compared to the standard implementation addmul_hmatrix, this algorithm uses H-matrix accumulators to reduce the number of singular value decompositions.

Parameters

alpha	Scaling factor $\alpha$ .
xtrans	Set if is to be used instead of .
x	Hierarchical matrix .
ytrans	Set if is to be used instead of .
y	Hierarchical matrix .
tm	Truncation mode.
eps	Truncation accuracy $\epsilon$ .
z	Target matrix .

void addmul_hmatrix	(	field	alpha,
		bool	xtrans,
		pchmatrix	x,
		bool	ytrans,
		pchmatrix	y,
		pctruncmode	tm,
		real	eps,
		phmatrix	z
	)

Multiply two H-matrices, $Z \gets \operatorname{succtrunc}(Z + \alpha X Y,\epsilon)$ .

Parameters

alpha	Scaling factor $\alpha$ .
xtrans	Set if is to be used instead of .
x	Hierarchical matrix .
ytrans	Set if is to be used instead of .
y	Hierarchical matrix .
tm	Truncation mode.
eps	Truncation accuracy $\epsilon$ .
z	Target matrix .

void addmul_hmatrix_amatrix_amatrix	(	field	alpha,
		bool	atrans,
		pchmatrix	a,
		bool	btrans,
		pcamatrix	bp,
		bool	ctrans,
		pamatrix	cp
	)

Multiply a matrix by a hmatrix, $C \gets C + \alpha A B$ .

Remarks: The matrices and are assumed to be ordered matching the row and column cluster trees of , respectively.

Parameters

alpha	Scaling factor $\alpha$ .
atrans	Set if is to be used instead of .
a	Hierarchical matrix .
btrans	Set if is to be used instead of .
bp	Matrix .
ctrans	Set if is to be used instead of .
cp	Target matrix .

void addmul_lower_hmatrix	(	field	alpha,
		bool	xtrans,
		pchmatrix	x,
		bool	ytrans,
		pchmatrix	y,
		pctruncmode	tm,
		real	eps,
		phmatrix	z
	)

Multiply two H-matrices, computing only the lower triangular part of the result, $\operatorname{lower}(Z) \gets \operatorname{succtrunc}(\operatorname{lower}(Z + \alpha X Y),\epsilon)$ .

Parameters

alpha	Scaling factor $\alpha$ .
xtrans	Set if is to be used instead of .
x	Hierarchical matrix .
ytrans	Set if is to be used instead of .
y	Hierarchical matrix .
tm	Truncation mode.
eps	Truncation accuracy $\epsilon$ .
z	Target matrix .

void addmul_rkmatrix_amatrix_amatrix	(	field	alpha,
		bool	atrans,
		pcrkmatrix	a,
		bool	btrans,
		pcamatrix	b,
		bool	ctrans,
		pamatrix	c
	)

Multiply a matrix by an rkmatrix, $C \gets C + \alpha A B$ .

Parameters

alpha	Scaling factor $\alpha$ .
atrans	Set if is to be used instead of .
a	Low-rank matrix .
btrans	Set if is to be used instead of .
b	Matrix .
ctrans	Set if is to be used instead of .
c	Target matrix .

void addproduct_haccum	(	field	alpha,
		bool	xtrans,
		pchmatrix	x,
		bool	ytrans,
		pchmatrix	y,
		phaccum	za
	)

Add a product $\alpha X Y$ to an accumulator.

Parameters

alpha	Scaling factor.
xtrans	Set if is to be used instead of .
x	H-matrix .
ytrans	Set if is to be used instead of .
y	H-matrix .
za	Accumulator for the result.

void choldecomp2_hmatrix	(	phmatrix	a,
		pctruncmode	tm,
		real	eps
	)

Compute the Cholesky factorization using accumulators, $A \approx L L^*$ .

The lower triangular part of $L$ is stored in the lower triangular part of the source matrix.

The strictly upper triangular part of the source matrix is not used.

Parameters

a	Source matrix , lower triangular part will be overwritten by .
tm	Truncation mode.
eps	Truncation accuracy.

void choldecomp_haccum ( phaccum aa )

Compute the Cholesky factorization using accumulators, $A \approx L L^*$ .

The lower triangular part of $L$ is stored in the lower triangular part of the source matrix.

The strictly upper triangular part of the source matrix is not used.

Parameters

aa	Accumulator representation of , lower triangular part will be overwritten

void choldecomp_hmatrix	(	phmatrix	a,
		pctruncmode	tm,
		real	eps
	)

Compute the Cholesky factorization, $A \approx L L^*$ .

The lower triangular part of $L$ is stored in the lower triangular part of the source matrix.

The strictly upper triangular part of the source matrix is not used.

Parameters

a	Source matrix , lower triangular part will be overwritten by .
tm	Truncation mode.
eps	Truncation accuracy.

void choleval_hmatrix_avector	(	pchmatrix	a,
		pavector	x
	)

Evaluate the linear system $x \gets A x$ using the Cholesky factorization provided by choldecomp_hmatrix.

Parameters

a	Matrix containing the Cholesky factorization in the form returned by choldecomp_hmatrix.
x	Right-hand side vector , that gets overwritten by the result.

void cholsolve_hmatrix_avector	(	pchmatrix	a,
		pavector	x
	)

Solve the linear system $A x = b$ using the Cholesky factorization provided by choldecomp_hmatrix.

Parameters

a	Matrix containing the Cholesky factorization in the form returned by choldecomp_hmatrix.
x	Right-hand side vector , will be overwritten by solution vector .

phmatrix clone_lower_hmatrix	(	bool	aunit,
		pchmatrix	a
	)

Create a duplicate of the lower triangular part of an H-matrix.

Parameters

aunit	Set if the duplicate has to have unit diagonal.
a	Source matrix .

Returns: Lower triangular matrix containing the lower triangular part of , with unit diagonal if aunit==true.

phmatrix clone_upper_hmatrix	(	bool	aunit,
		pchmatrix	a
	)

Create a duplicate of the upper triangular part of an H-matrix.

Parameters

aunit	Set if the duplicate has to have unit diagonal.
a	Source matrix .

Returns: Upper triangular matrix containing the upper triangular part of , with unit diagonal if aunit==true.

void del_haccum ( phaccum ha )

Delete an accumulator.

Parameters

ha	Accumulator to be deleted.

void flush_haccum ( phaccum ha )

Flush an H-matrix accumulator, i.e., add all accumulated products to the target matrix and clear the accumulator.

Parameters

ha	H-matrix accumulator.

void invert_hmatrix	(	phmatrix	a,
		phmatrix	work,
		pctruncmode	tm,
		real	eps
	)

Invert an H-matrix, $A \gets \operatorname{succtrunc}(A^{-1},\epsilon)$ .

Parameters

a	Matrix , gets overwritten with $A^{-1}$ .
work	Auxiliary storage, has to be zero and be of the same structure as `a`. Consider using clonestructure_hmatrix.
tm	Truncation mode.
eps	Truncation accuracy.

void lowersolve2_hmatrix_hmatrix	(	bool	aunit,
		bool	atrans,
		pchmatrix	a,
		pctruncmode	tm,
		real	eps,
		bool	xtrans,
		phmatrix	x
	)

Solve a lower triangular system using accumulators, $X \gets \operatorname{succtrunc}(L^{-1} X, \epsilon)$ or $X \gets \operatorname{succtrunc}(L^{-*} X, \epsilon)$ .

Parameters

aunit	Set if has unit diagonal.
atrans	Set if instead of is to be used.
a	Matrix containing the lower triangular part of . The remainder is not used.
tm	Truncation mode.
eps	Truncation accuracy.
xtrans	set if instead of is to be used.
x	Input matrix , will be overwritten by the solution. The rows are assumed to be in cluster numbering according to `a->rc`.

void lowersolve_haccum	(	bool	aunit,
		bool	atrans,
		pchmatrix	a,
		bool	xtrans,
		phaccum	xa
	)

Solve a lower triangular system using accumulators, $X \gets \operatorname{succtrunc}(L^{-1} X, \epsilon)$ or $X \gets \operatorname{succtrunc}(L^{-*} X, \epsilon)$ .

Parameters

aunit	Set if has unit diagonal.
atrans	Set if instead of is to be used.
a	Matrix containing the lower triangular part of . The remainder is not used.
xtrans	set if instead of is to be used.
xa	Accumulator representation of , will be overwritten by the solution.

void lrdecomp2_hmatrix	(	phmatrix	a,
		pctruncmode	tm,
		real	eps
	)

Compute the LR factorization using accumulators, $A \approx L R$ .

The lower triangular part $L$ has unit diagonal, only its strict lower triangular part is stored in the strict lower triangular part of the source matrix.

The upper triangular part is stored in the upper triangular part of the source matrix.

Parameters

a	Source matrix , will be overwritten with and .
tm	Truncation mode.
eps	Truncation accuracy.

void lrdecomp_haccum ( phaccum aa )

Compute the LR factorization using accumulators, $A \approx L R$ .

The lower triangular part $L$ has unit diagonal, only its strict lower triangular part is stored in the strict lower triangular part of the source matrix.

The upper triangular part is stored in the upper triangular part of the source matrix.

Parameters

aa	Accumulator representation of , will be overwritten with and .

void lrdecomp_hmatrix	(	phmatrix	a,
		pctruncmode	tm,
		real	eps
	)

Compute the LR factorization, $A \approx L R$ .

The lower triangular part $L$ has unit diagonal, only its strict lower triangular part is stored in the strict lower triangular part of the source matrix.

The upper triangular part is stored in the upper triangular part of the source matrix.

Parameters

a	Source matrix , will be overwritten by and .
tm	Truncation mode.
eps	Truncation accuracy.

void lreval_hmatrix_avector	(	bool	atrans,
		pchmatrix	a,
		pavector	x
	)

Evaluate the linear system $x \gets A x$ or $x \gets A^* x$ using the LR factorization provided by lrdecomp_hmatrix.

Parameters

atrans	Set if instead of is to be used.
a	Matrix containing the LR factorization in the form returned by lrdecomp_hmatrix.
x	Right-hand side vector , that gets overwritten by the result.

void lreval_n_hmatrix_avector	(	pchmatrix	a,
		pavector	x
	)

Evaluate the linear system $x \gets A x$ using the LR factorization provided by lrdecomp_hmatrix.

Parameters

a	Matrix containing the LR factorization in the form returned by lrdecomp_hmatrix.
x	Right-hand side vector , that gets overwritten by the result.

void lreval_t_hmatrix_avector	(	pchmatrix	a,
		pavector	x
	)

Evaluate the linear system $x \gets A^* x$ using the LR factorization provided by lrdecomp_hmatrix.

Parameters

a	Matrix containing the LR factorization in the form returned by lrdecomp_hmatrix.
x	Right-hand side vector , that gets overwritten by the result.

void lrsolve_hmatrix_avector	(	bool	atrans,
		pchmatrix	a,
		pavector	x
	)

Solve the linear systems $A x = b$ or $A^* x = b$ using the LR factorization provided by lrdecomp_hmatrix.

Parameters

atrans	Set if instead of is to be used.
a	Matrix containing the LR factorization in the form returned by lrdecomp_hmatrix.
x	Right-hand side vector , will be overwritten by solution vector .

void lrsolve_n_hmatrix_avector	(	pchmatrix	a,
		pavector	x
	)

Solve the linear systems $A x = b$ using the LR factorization provided by lrdecomp_hmatrix.

Parameters

a	Matrix containing the LR factorization in the form returned by lrdecomp_hmatrix.
x	Right-hand side vector , will be overwritten by solution vector .

void lrsolve_t_hmatrix_avector	(	pchmatrix	a,
		pavector	x
	)

Solve the linear systems $A^* x = b$ using the LR factorization provided by lrdecomp_hmatrix.

Parameters

a	Matrix containing the LR factorization in the form returned by lrdecomp_hmatrix.
x	Right-hand side vector , will be overwritten by solution vector .

prkmatrix merge_hmatrix_rkmatrix	(	pchmatrix	s,
		pctruncmode	tm,
		real	eps
	)

Merge a depth-one hmatrix containing only rkmatrix submatrices into a new rkmatrix, $A \gets \operatorname{succtrunc}\left(\begin{pmatrix} B_{11} & \ldots & B_{1m}\\ \vdots & \ddots & \vdots\\ B_{n1} & \ldots & B_{nm} \end{pmatrix},\epsilon\right)$ .

Parameters

s	Source matrix
tm	Truncation mode.
eps	Truncation accuracy $\epsilon$ .

Returns: rkmatrix approximating the block matrix.

void merge_rkmatrix	(	bool	colmerge,
		pcrkmatrix	src,
		pctruncmode	tm,
		real	eps,
		prkmatrix	trg
	)

Split a leaf hmatrix into submatrices.

** Split a leaf hmatrix into submatrices referencing Block-merge two rk matrices into a new rk matrix, $A \gets \operatorname{trunc}\left(\begin{pmatrix} A & B \end{pmatrix},\epsilon\right)$ or $A \gets \operatorname{trunc}\left(\begin{pmatrix} A\\ B \end{pmatrix},\epsilon\right)$ .

If colmerge is not set, $A$ and $B$ have to have the same number of rows, otherwise the same number of columns.

Attention: The matrix is the last parameter, although it is placed in the first row or column.

Parameters

colmerge	Set to merge into a column instead of a row.
src	Source matrix .
tm	Truncation mode.
eps	Truncation accuracy $\epsilon$ .
trg	Target matrix , will be overwritten by result.

phaccum new_haccum	(	phmatrix	z,
		pctruncmode	tm,
		real	eps
	)

Create a new H-matrix accumulator.

Parameters

z	Target matrix.
tm	Truncation mode.
eps	Truncation accuracy.

Returns: Accumulator for z

phaccum* split_haccum	(	phmatrix	z,
		phaccum	za
	)

Create H-matrix accumulators for submatrices.

Parameters

z	Target matrix, row and column cluster have to conincide with `ha->z->rc` and `ha->z->cc`.
za	H-matrix accumulator.

Returns: Array containing accumulators for the submatrices of z, numbered as in z->son.

phmatrix split_rkmatrix	(	pcrkmatrix	r,
		pccluster	rc,
		pccluster	cc,
		bool	rsplit,
		bool	csplit,
		bool	copy
	)

Split an rkmatrix into submatrices.

Parameters

r	Original matrix.
rc	Row cluster.
cc	Column cluster.
rsplit	Set to split in row direction.
csplit	Set to split in column direction.
copy	Set to copy values into new matrix, otherwise it will be zero.

Returns: Block matrix of depth one.

phmatrix split_sub_amatrix	(	pcamatrix	f,
		pccluster	rc,
		pccluster	cc,
		bool	rsplit,
		bool	csplit
	)

Split an amatrix into submatrices.

*Split an amatrix into submatrices referencing the original matrix.

Parameters

f	Original matrix.
rc	Row cluster.
cc	Column cluster.
rsplit	Set to split in row direction.
csplit	Set to split in column direction.

Returns: Block matrix of depth one.

phmatrix split_sub_rkmatrix	(	pcrkmatrix	r,
		pccluster	rc,
		pccluster	cc,
		bool	rsplit,
		bool	csplit
	)

Split an rkmatrix into submatrices referencing the original matrix.

Attention: Since the blocks of the result contain only references to the original matrix, they cannot be resized, and changing them changes the other submatrices in the same row and column. It's probably best to treat the submatrices as constant.

Parameters

r	Original matrix.
rc	Row cluster.
cc	Column cluster.
rsplit	Set to split in row direction.
csplit	Set to split in column direction.

Returns: Block matrix of depth one.

void triangulareval_hmatrix_avector	(	bool	alower,
		bool	aunit,
		bool	atrans,
		pchmatrix	a,
		pavector	x
	)

Evaluate a triangular system $x \gets A x$ or $x \gets A^* x$ .

Parameters

alower	Set if is lower triangular, otherwise it is assumed to be upper triangular.
aunit	Set if has unit diagonal.
atrans	Set if instead of is to be used.
a	Matrix containing the lower or upper triangular part of . The remainder of the matrix is not used.
x	Input vector , will be overwritten by the solution.

void triangularinvmul_amatrix_hmatrix	(	bool	alower,
		bool	aunit,
		bool	atrans,
		pcamatrix	a,
		pctruncmode	tm,
		real	eps,
		bool	xtrans,
		phmatrix	xp
	)

Solve a triangular system, $X \gets \operatorname{succtrunc}(A^{-1} X, \epsilon)$ or $X \gets \operatorname{succtrunc}(A^{-*} X, \epsilon)$ .

Parameters

alower	Set if is lower triangular, otherwise it is assumed to by upper triangular.
aunit	Set if has unit diagonal.
atrans	Set if instead of is to be used.
a	Matrix containing the lower or upper triangular part of . The remainder of the matrix is not used.
tm	Truncation mode.
eps	Truncation accuracy.
xtrans	set if instead of is to be used.
xp	Input matrix , will be overwritten by the solution. The rows are assumed to be in cluster numbering according to `a->rc`.

void triangularinvmul_hmatrix	(	bool	alower,
		bool	aunit,
		bool	atrans,
		pchmatrix	a,
		pctruncmode	tm,
		real	eps,
		bool	xtrans,
		phmatrix	xp
	)

Solve a triangular system, $X \gets \operatorname{succtrunc}(A^{-1} X, \epsilon)$ or $X \gets \operatorname{succtrunc}(A^{-*} X, \epsilon)$ .

Parameters

alower	Set if is lower triangular, otherwise it is assumed to be upper triangular.
aunit	Set if has unit diagonal.
atrans	Set if instead of is to be used.
a	Matrix containing the triangular part of . The remainder is not used.
tm	Truncation mode.
eps	Truncation accuracy.
xtrans	set if instead of is to be used.
xp	Input matrix , will be overwritten by the solution. The rows are assumed to be in cluster numbering according to `a->rc`.

void triangularinvmul_hmatrix_amatrix	(	bool	alower,
		bool	aunit,
		bool	atrans,
		pchmatrix	a,
		bool	xtrans,
		pamatrix	xp
	)

Solve a triangular system $X \gets A^{-1} X$ or $X \gets (A^*)^{-1} X$ .

Parameters

alower	Set if is lower triangular, otherwise it is assumed to by upper triangular.
aunit	Set if has unit diagonal.
atrans	Set if instead of is to be used.
a	Matrix containing the lower or upper triangular part of . The remainder of the matrix is not used.
xtrans	set if instead of is to be used.
xp	Input matrix , will be overwritten by the solution. The rows are assumed to be in cluster numbering according to `a->rc`.

void triangularinvmul_hmatrix_avector	(	bool	alower,
		bool	aunit,
		bool	atrans,
		pchmatrix	a,
		pavector	xp
	)

Solve a triangular system, $x \gets R^{-1} x$ or $x \gets R^{-*} x$ .

Parameters

alower	Set if is lower triangular, otherwise it is assumed to be upper triangular.
aunit	Set if has unit diagonal.
atrans	Set if instead of is to be used.
a	Matrix containing the upper triangular part of . The strictly lower triangular part is not used.
xp	Input vector , will be overwritten by the solution. The vector is assumed to be in cluster numbering according to `a->cc` respectively `a->rc`.

void triangularmul_hmatrix	(	bool	alower,
		bool	aunit,
		bool	atrans,
		pchmatrix	a,
		pctruncmode	tm,
		real	eps,
		bool	xtrans,
		phmatrix	xp
	)

Evaluate a triangular system, $X \gets \operatorname{succtrunc}(A X, \epsilon)$ or $X \gets \operatorname{succtrunc}(A^{*} X, \epsilon)$ .

Parameters

alower	Set if is lower triangular, otherwise it is assumed to by upper triangular.
aunit	Set if has unit diagonal.
atrans	Set if instead of is to be used.
a	Matrix containing the lower or upper triangular part of . The remainder of the matrix is not used.
tm	Truncation mode.
eps	Truncation accuracy.
xtrans	set if instead of is to be used.
xp	Input matrix , will be overwritten by the solution. The rows are assumed to be in cluster numbering according to `a->rc`.

void triangularmul_hmatrix_amatrix	(	bool	alower,
		bool	aunit,
		bool	atrans,
		pchmatrix	a,
		bool	xtrans,
		pamatrix	xp
	)

Evaluate a triangular system $X \gets A X$ or $X \gets A^* X$ .

Parameters

alower	Set if is lower triangular, otherwise it is assumed to by upper triangular.
aunit	Set if has unit diagonal.
atrans	Set if instead of is to be used.
a	Matrix containing the lower or upper triangular part of . The remainder of the matrix is not used.
xtrans	set if instead of is to be used.
xp	Input matrix , will be overwritten by the solution. The rows are assumed to be in cluster numbering according to `a->rc`.

void triangularmul_hmatrix_avector	(	bool	alower,
		bool	aunit,
		bool	atrans,
		pchmatrix	a,
		pavector	xp
	)

Evaluate a triangular system $x \gets A x$ or $x \gets A^* x$ .

Parameters

alower	Set if is lower triangular, otherwise it is assumed to be upper triangular.
aunit	Set if has unit diagonal.
atrans	Set if instead of is to be used.
a	Matrix containing the lower or upper triangular part of . The remainder of the matrix is not used.
xp	Input vector , will be overwritten by the solution. The rows are assumed to be in cluster numbering according to `a->rc`.

void triangularsolve_hmatrix_avector	(	bool	alower,
		bool	aunit,
		bool	atrans,
		pchmatrix	a,
		pavector	x
	)

Solve a triangular system $x \gets A^{-1} x$ or $x \gets (A^*)^{-1} x$ .

Parameters

alower	Set if is lower triangular, otherwise it is assumed to be upper triangular.
aunit	Set if has unit diagonal.
atrans	Set if instead of is to be used.
a	Matrix containing the lower or upper triangular part of . The remainder of the matrix is not used.
x	Input vector , will be overwritten by the solution.

void trunc_rkmatrix	(	pctruncmode	tm,
		real	eps,
		prkmatrix	r
	)

Truncate an rkmatrix, $A \gets \operatorname{trunc}(A,\epsilon)$ .

Parameters

tm	Truncation mode.
eps	Truncation accuracy $\epsilon$ .
r	Source matrix, will be overwritten by truncated matrix.

void uppersolve2_hmatrix_hmatrix	(	bool	aunit,
		bool	atrans,
		pchmatrix	a,
		pctruncmode	tm,
		real	eps,
		bool	xtrans,
		phmatrix	x
	)

Solve an upper triangular system using accumulators, $X \gets \operatorname{succtrunc}(R^{-1} X, \epsilon)$ or $X \gets \operatorname{succtrunc}(R^{-*} X, \epsilon)$ .

Parameters

aunit	Set if has unit diagonal.
atrans	Set if instead of is to be used.
a	Matrix containing the lower triangular part of . The remainder is not used.
tm	Truncation mode.
eps	Truncation accuracy.
xtrans	set if instead of is to be used.
x	Input matrix , will be overwritten by the solution. The rows are assumed to be in cluster numbering according to `a->rc`.

void uppersolve_haccum	(	bool	aunit,
		bool	atrans,
		pchmatrix	a,
		bool	xtrans,
		phaccum	xa
	)

Solve an upper triangular system using accumulators, $X \gets \operatorname{succtrunc}(R^{-1} X, \epsilon)$ or $X \gets \operatorname{succtrunc}(R^{-*} X, \epsilon)$ .

Parameters

aunit	Set if has unit diagonal.
atrans	Set if instead of is to be used.
a	Matrix containing the upper triangular part of . The remainder is not used.
xtrans	set if instead of is to be used.
xa	Accumulator representation of , will be overwritten by the solution.

Data Structures

Typedefs

Functions

Detailed Description

Function Documentation