Iterative solvers of Krylov type. More...

Typedefs
typedef void(*	addeval_t) (field alpha, void *matrix, pcavector x, pavector y)
	Matrix callback. More...

typedef void(*	mvm_t) (field alpha, bool trans, void *matrix, pcavector x, pavector y)
	Matrix callback. More...

typedef void(*	prcd_t) (void *pdata, pavector r)
	Preconditioner callback. More...

Functions
real	norm2_matrix (mvm_t mvm, void *A, uint rows, uint cols)
	Approximate the spectral norm $\\|A\\|_2$ of a matrix . More...

real	norm2diff_matrix (mvm_t mvmA, void A, mvm_t mvmB, void B, uint rows, uint cols)
	Approximate the spectral norm $\\|A-B\\|_2$ of the difference of two matrices and . More...

real	norm2diff_pre_matrix (mvm_t mvmA, void A, prcd_t evalB, prcd_t evaltransB, void B, uint rows, uint cols)
	Approximate the spectral norm $\\|A-B\\|_2$ of the difference of two matrices and . The matrix is given as a factorization and can be applied to some vector. More...

real	norm2diff_id_pre_matrix (mvm_t mvmA, void A, prcd_t solveB, prcd_t solvetransB, void B, uint rows, uint cols)
	Approximate the spectral norm $\\|I - B^{-1}A\\|_2$ . The matrix is given as a factorization and can be applied to some vector. Usually is an approximation to and therefore this functions measures the quality of some preconditioner for . More...

void	init_cg (addeval_t addeval, void *matrix, pcavector b, pavector x, pavector r, pavector p, pavector a)
	Initialize a standard conjugate gradient method to solve . More...

void	step_cg (addeval_t addeval, void *matrix, pcavector b, pavector x, pavector r, pavector p, pavector a)
	One step of a standard conjugate gradient method. More...

real	evalfunctional_cg (addeval_t addeval, void *matrix, pcavector b, pcavector x, pcavector r)
	error information of a standard conjugate gradient method More...

void	init_pcg (addeval_t addeval, void matrix, prcd_t prcd, void pdata, pcavector b, pavector x, pavector r, pavector q, pavector p, pavector a)
	Initialize a preconditioned conjugate gradient method to solve $N^{1/2} A N^{1/2} \widehat{x} = N^{1/2} b$ with $x = N^{1/2} \widehat{x}$ . More...

void	step_pcg (addeval_t addeval, void matrix, prcd_t prcd, void pdata, pcavector b, pavector x, pavector r, pavector q, pavector p, pavector a)
	One step of a preconditioned conjugate gradient method. More...

void	init_uzawa (prcd_t solve_A11, void matrix_A11, mvm_t mvm_A21, void matrix_A21, pcavector b1, pcavector b2, pavector x1, pavector x2, pavector r2, pavector p2, pavector a1, pavector s2)
	Initialize the standard Uzawa iteration. More...

void	step_uzawa (prcd_t solve_A11, void matrix_A11, mvm_t mvm_A21, void matrix_A21, pcavector b1, pcavector b2, pavector x1, pavector x2, pavector r2, pavector p2, pavector a1, pavector s2)
	Perform one step of the standard Uzawa iteration. More...

void	init_puzawa (prcd_t solve_A11, void matrix_A11, mvm_t mvm_A21, void matrix_A21, prcd_t prcd, void *pdata, pcavector b1, pcavector b2, pavector x1, pavector x2, pavector r2, pavector q2, pavector p2, pavector a1, pavector s2)
	Initialize a preconditioned Uzawa iteration. More...

void	step_puzawa (prcd_t solve_A11, void matrix_A11, mvm_t mvm_A21, void matrix_A21, prcd_t prcd, void *pdata, pcavector b1, pcavector b2, pavector x1, pavector x2, pavector r2, pavector q2, pavector p2, pavector a1, pavector s2)
	Perform one step of the preconditioned Uzawa iteration. More...

void	init_bicg (addeval_t addeval, addeval_t addevaltrans, void *matrix, pcavector b, pavector x, pavector r, pavector rt, pavector p, pavector pt, pavector a, pavector at)
	Initialize a biconjugate gradient method. More...

void	step_bicg (addeval_t addeval, addeval_t addevaltrans, void *matrix, pcavector b, pavector x, pavector r, pavector rt, pavector p, pavector pt, pavector a, pavector at)
	One step a biconjugate gradient method. More...

void	init_bicgstab (addeval_t addeval, void *matrix, pcavector b, pavector x, pavector r, pavector rt, pavector p, pavector a, pavector as)
	Initialize a stabilized biconjugate gradient method. More...

void	step_bicgstab (addeval_t addeval, void *matrix, pcavector b, pavector x, pavector r, pavector rt, pavector p, pavector a, pavector as)
	One step a stabilized biconjugate gradient method. More...

void	init_gmres (addeval_t addeval, void matrix, pcavector b, pavector x, pavector rhat, pavector q, uint kk, pamatrix qr, pavector tau)
	Initialize GMRES. More...

void	step_gmres (addeval_t addeval, void matrix, pcavector b, pavector x, pavector rhat, pavector q, uint kk, pamatrix qr, pavector tau)
	One step of the GMRES method. More...

void	finish_gmres (addeval_t addeval, void matrix, pcavector b, pavector x, pavector rhat, pavector q, uint kk, pamatrix qr, pavector tau)
	Completes or restarts the GMRES method. More...

real	residualnorm_gmres (pcavector rhat, uint k)
	Returns norm of current residual vector. More...

void	init_pgmres (addeval_t addeval, void matrix, prcd_t prcd, void pdata, pcavector b, pavector x, pavector rhat, pavector q, uint *kk, pamatrix qr, pavector tau)
	Initialize preconditioned GMRES. More...

void	step_pgmres (addeval_t addeval, void matrix, prcd_t prcd, void pdata, pcavector b, pavector x, pavector rhat, pavector q, uint *kk, pamatrix qr, pavector tau)
	One step of the preconditioned GMRES method. More...

void	finish_pgmres (addeval_t addeval, void matrix, prcd_t prcd, void pdata, pcavector b, pavector x, pavector rhat, pavector q, uint *kk, pamatrix qr, pavector tau)
	Completes or restarts the preconditioned GMRES method. More...

real	residualnorm_pgmres (pcavector rhat, uint k)
	Returns norm of current preconditioned residual vector. More...

Detailed Description

Iterative solvers of Krylov type.

Typedef Documentation

typedef void(* addeval_t) (field alpha, void *matrix, pcavector x, pavector y)

Matrix callback.

Used to evaluate the system matrix $A$ or its adjoint, i.e., to perform $y \gets y + \alpha A x$ .

Functions like addeval_amatrix_avector, addevaltrans_amatrix_avector, addeval_hmatrix_avector, addevaltrans_hmatrix_avector, addeval_h2matrix_avector, addevaltrans_h2matrix_avector or addeval_sparsematrix_avector can be cast to addeval_t.

Parameters

alpha	Scaling factor $\alpha$ .
matrix	Matrix data describing .
x	Source vector .
y	Target vector .

typedef void(* mvm_t) (field alpha, bool trans, void *matrix, pcavector x, pavector y)

Matrix callback.

Used to evaluate the system matrix $A$ or its adjoint, i.e., to perform $y \gets y + \alpha A x$ .

Compared to addeval_t, callbacks of this type allow us to evaluate both the matrix and its adjoint.

Functions like mvm_amatrix_avector, mvm_hmatrix_avector, mvm_h2matrix_avector, or mvm_sparsematrix_avector can be cast to mvm_t.

Parameters

alpha	Scaling factor $\alpha$ .
trans	Set if is to be used instead of .
matrix	Matrix data describing .
x	Source vector .
y	Target vector .

typedef void(* prcd_t) (void *pdata, pavector r)

Preconditioner callback.

Used to apply a precondtioner to a vector, i.e., to perform $r \gets N r$ .

Functions like lrsolve_amatrix_avector, cholsolve_amatrix_avector, ldltsolve_amatrix_avector, qrsolve_amatrix_avector, lrsolve_hmatrix_avector or cholsolve_hmatrix_avector can be cast to prcd_t.

Parameters

pdata	Preconditioner data describing .
r	Source vector, will be overwritten by result.

Function Documentation

real evalfunctional_cg	(	addeval_t	addeval,
		void *	matrix,
		pcavector	b,
		pcavector	x,
		pcavector	r
	)

error information of a standard conjugate gradient method

Parameters

addeval	Callback function name
matrix	untyped pointer to matrix data (A)
b	Right-hand side.
x	Approximate solution.
r	Residual

Returns

void finish_gmres	(	addeval_t	addeval,
		void *	matrix,
		pcavector	b,
		pavector	x,
		pavector	rhat,
		pavector	q,
		uint *	kk,
		pamatrix	qr,
		pavector	tau
	)

Completes or restarts the GMRES method.

Solves the least-squares problem $Q_{k+1}^* A Q_k \widehat{x} = Q_{k+1}^* r$ and performs the update $x \gets x + Q_k \widehat{x}$ .

The function calls init_gmres to reset the iteration and prepare for a restart.

Parameters

addeval	Callback function representing the matrix .
matrix	Data for the `addeval` callback.
b	Right-hand side vector .
x	Initial guess for the solution , will eventually be replaced by an improved approximation.
rhat	Transformed residual. The absolute value of `rhat[*kk]` is the Euclidean norm of the residual.
q	Next vector of the Krylov basis, constructed by Householder's elementary reflectors.
kk	Pointer to current dimension of the Krylov space.
qr	Representation of the Arnoldi basis $Q_{k+1}$ and the transformed matrix $Q_{k+1}^* A Q_k$ .
tau	Scaling factors of elementary reflectors, provided by qrdecomp_amatrix.

void finish_pgmres	(	addeval_t	addeval,
		void *	matrix,
		prcd_t	prcd,
		void *	pdata,
		pcavector	b,
		pavector	x,
		pavector	rhat,
		pavector	q,
		uint *	kk,
		pamatrix	qr,
		pavector	tau
	)

Completes or restarts the preconditioned GMRES method.

Solves the least-squares problem $Q_{k+1}^* N A Q_k \widehat{x} = Q_{k+1}^* N r$ and performs the update $x \gets x + Q_k \widehat{x}$ .

The function calls init_pgmres to reset the iteration and prepare for a restart.

Parameters

addeval	Callback function representing the matrix .
matrix	Data for the `addeval` callback.
prcd	Callback function representing the preconditioner .
pdata	Data for the `prcd` callback.
b	Right-hand side vector .
x	Initial guess for the solution , will eventually be replaced by an improved approximation.
rhat	Transformed residual. The absolute value of `rhat[*kk]` is the Euclidean norm of the preconditioned residual.
q	Next vector of the Krylov basis, constructed by Householder's elementary reflectors.
kk	Pointer to current dimension of the Krylov space.
qr	Representation of the Arnoldi basis $Q_{k+1}$ and the transformed matrix $Q_{k+1}^* A Q_k$ .
tau	Scaling factors of elementary reflectors, provided by qrdecomp_amatrix.

void init_bicg	(	addeval_t	addeval,
		addeval_t	addevaltrans,
		void *	matrix,
		pcavector	b,
		pavector	x,
		pavector	r,
		pavector	rt,
		pavector	p,
		pavector	pt,
		pavector	a,
		pavector	at
	)

Initialize a biconjugate gradient method.

Parameters

addeval	Callback function name
addevaltrans	Callback function name
matrix	untyped pointer to matrix data
b	Right-hand side.
x	Approximate solution.
r	Residual b-Ax
rt	Adjoint residual
p	Search direction
pt	Adjoint search direction
a	auxiliary vector
at	auxiliary vector

void init_bicgstab	(	addeval_t	addeval,
		void *	matrix,
		pcavector	b,
		pavector	x,
		pavector	r,
		pavector	rt,
		pavector	p,
		pavector	a,
		pavector	as
	)

Initialize a stabilized biconjugate gradient method.

Parameters

addeval	Callback function name
matrix	untyped pointer to matrix data
b	Right-hand side.
x	Approximate solution.
r	Residual b-Ax
rt	Adjoint residual
p	Search direction
a	auxiliary vector
as	auxiliary vector

void init_cg	(	addeval_t	addeval,
		void *	matrix,
		pcavector	b,
		pavector	x,
		pavector	r,
		pavector	p,
		pavector	a
	)

Initialize a standard conjugate gradient method to solve $A x = b$ .

The matrix $A$ has to be self-adjoint and positive definite (or at least positive semidefinite).

Parameters

addeval	Callback function name
matrix	untyped pointer to matrix data
b	Right-hand side.
x	Approximate solution.
r	Residual .
p	Search direction.
a	Auxiliary vector.

void init_gmres	(	addeval_t	addeval,
		void *	matrix,
		pcavector	b,
		pavector	x,
		pavector	rhat,
		pavector	q,
		uint *	kk,
		pamatrix	qr,
		pavector	tau
	)

Initialize GMRES.

The parameters are prepared for solving $A x = b$ with the GMRES method.

The maximal dimension of the Krylov subspace is determined by the number of columns of qr: for a k-dimensional subspace, qr->cols==k+1 is required.

Parameters

addeval	Callback function representing the matrix .
matrix	Data for the `addeval` callback.
b	Right-hand side vector .
x	Initial guess for the solution , will eventually be replaced by an improved approximation.
rhat	Transformed residual. The absolute value of `rhat[*kk]` is the Euclidean norm of the residual.
q	Next vector of the Krylov basis, constructed by Householder's elementary reflectors.
kk	Pointer to current dimension of the Krylov space.
qr	Representation of the Arnoldi basis $Q_{k+1}$ and the transformed matrix $Q_{k+1}^* A Q_k$ .
tau	Scaling factors of elementary reflectors, provided by qrdecomp_amatrix.

void init_pcg	(	addeval_t	addeval,
		void *	matrix,
		prcd_t	prcd,
		void *	pdata,
		pcavector	b,
		pavector	x,
		pavector	r,
		pavector	q,
		pavector	p,
		pavector	a
	)

Initialize a preconditioned conjugate gradient method to solve $N^{1/2} A N^{1/2} \widehat{x} = N^{1/2} b$ with $x = N^{1/2} \widehat{x}$ .

The matrix $A$ has to be self-adjoint and positive definite (or at least positive semidefinite). The matrix $N$ has to be self-adjoint and positive definite.

Parameters

addeval	Callback function name
matrix	untyped pointer to matrix data
prcd	Callback function name
pdata	untyped pointer to matrix data
b	Right-hand side.
x	Approximate solution.
r	Residual b-Ax
q	Preconditioned residual
p	Search direction
a	auxiliary vector

void init_pgmres	(	addeval_t	addeval,
		void *	matrix,
		prcd_t	prcd,
		void *	pdata,
		pcavector	b,
		pavector	x,
		pavector	rhat,
		pavector	q,
		uint *	kk,
		pamatrix	qr,
		pavector	tau
	)

Initialize preconditioned GMRES.

The parameters are prepared for solving $N A x = N b$ with the GMRES method.

The maximal dimension of the Krylov subspace is determined by the number of columns of qr: for a k-dimensional subspace, qr->cols==k+1 is required.

Parameters

addeval	Callback function representing the matrix .
matrix	Data for the `addeval` callback.
prcd	Callback function representing the preconditioner .
pdata	Data for the `prcd` callback.
b	Right-hand side vector .
x	Initial guess for the solution , will eventually be replaced by an improved approximation.
rhat	Transformed residual. The absolute value of `rhat[*kk]` is the Euclidean norm of the preconditioned residual.
q	Next vector of the Krylov basis, constructed by Householder's elementary reflectors.
kk	Pointer to current dimension of the Krylov space.
qr	Representation of the Arnoldi basis $Q_{k+1}$ and the transformed matrix $Q_{k+1}^* A Q_k$ .
tau	Scaling factors of elementary reflectors, provided by qrdecomp_amatrix.

void init_puzawa	(	prcd_t	solve_A11,
		void *	matrix_A11,
		mvm_t	mvm_A21,
		void *	matrix_A21,
		prcd_t	prcd,
		void *	pdata,
		pcavector	b1,
		pcavector	b2,
		pavector	x1,
		pavector	x2,
		pavector	r2,
		pavector	q2,
		pavector	p2,
		pavector	a1,
		pavector	s2
	)

Initialize a preconditioned Uzawa iteration.

The preconditioned Uzawa iteration solves saddle point systems of the form

$\begin{pmatrix} A_{11} & A_{21}^*\\ A_{21} & \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix} = \begin{pmatrix} b_1\\ b_2 \end{pmatrix}$

by applying the preconditioned conjugate gradient method to the Schur complement. The matrix $A_{11}$ has to be symmetric and positive definite. If $A_{21}$ is not surjective, $b_2$ will only be determined up to an element of the null space of $A_{21}^*$ .

Parameters

solve_A11	Callback for solving $A_{11} x_1 = b_1$
matrix_A11	Data for `solve_A11` callback
mvm_A21	Callback for evaluating $A_{21} x_1$ or $A_{21}^* x_2$
matrix_A21	Data for `mvm_A21` callback
prcd	Callback function name
pdata	untyped pointer to matrix data
b1	Right-hand side vector
b2	Right-hand side vector
x1	Approximate solution vector
x2	Approximate solution vector
r2	Residual of the Schur complement system, may be useful for a stopping criterion
p2	Search direction for Schur complement system
q2	Auxiliary variable of the same dimension as
a1	Auxiliary variable of the same dimension as
s2	Auxiliary variable of the same dimension as

void init_uzawa	(	prcd_t	solve_A11,
		void *	matrix_A11,
		mvm_t	mvm_A21,
		void *	matrix_A21,
		pcavector	b1,
		pcavector	b2,
		pavector	x1,
		pavector	x2,
		pavector	r2,
		pavector	p2,
		pavector	a1,
		pavector	s2
	)

Initialize the standard Uzawa iteration.

The Uzawa iteration solves saddle point systems of the form

$\begin{pmatrix} A_{11} & A_{21}^*\\ A_{21} & \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix} = \begin{pmatrix} b_1\\ b_2 \end{pmatrix}$

by applying the conjugate gradient method to the Schur complement. The matrix $A_{11}$ has to be symmetric and positive definite. If $A_{21}$ is not surjective, $b_2$ will only be determined up to an element of the null space of $A_{21}^*$ .

Parameters

solve_A11	Callback for solving $A_{11} x_1 = b_1$
matrix_A11	Data for `solve_A11` callback
mvm_A21	Callback for evaluating $A_{21} x_1$ or $A_{21}^* x_2$
matrix_A21	Data for `mvm_A21` callback
b1	Right-hand side vector
b2	Right-hand side vector
x1	Approximate solution vector
x2	Approximate solution vector
r2	Residual of the Schur complement system, may be useful for a stopping criterion
p2	Search direction for Schur complement system
a1	Auxiliary variable of the same dimension as
s2	Auxiliary variable of the same dimension as

real norm2_matrix	(	mvm_t	mvm,
		void *	A,
		uint	rows,
		uint	cols
	)

Approximate the spectral norm $\|A\|_2$ of a matrix $A$ .

The spectral norm is approximated by applying a few steps of the power iteration to the matrix $A^* A$ and computing the square root of the resulting eigenvalue approximation.

Parameters

mvm	Callback function for evaluating `A` and its adjoint.
A	Matrix .
rows	Number of rows of `A`.
cols	Number of columns of `A`.

Returns: Approximation of $\|A\|_2$ .

real norm2diff_id_pre_matrix	(	mvm_t	mvmA,
		void *	A,
		prcd_t	solveB,
		prcd_t	solvetransB,
		void *	B,
		uint	rows,
		uint	cols
	)

Approximate the spectral norm $\|I - B^{-1}A\|_2$ . The matrix $B$ is given as a factorization and can be applied to some vector. Usually $B$ is an approximation to $A$ and therefore this functions measures the quality of some preconditioner for $A$ .

The spectral norm is approximated by applying a few steps of the power iteration to the matrix $(I - B^{-1}A)^* (I - B^{-1}A)$ and computing the square root of the resulting eigenvalue approximation. $B$ is usually given as a LR decomposition or cholesky factorization of $A$ .

Parameters

mvmA	Callback function for evaluating `A` and its adjoint.
A	Matrix .
solveB	Callback function for solving a linear system with `B`.
solvetransB	Callback function for solving a linear system with the adjoint of `B`.
B	Factorized matrix .
rows	Number of rows of either `A` and `B`.
cols	Number of columns of either `A` and `B`.

Returns: Approximation of $\|I - B^{-1}A\|_2$ .

real norm2diff_matrix	(	mvm_t	mvmA,
		void *	A,
		mvm_t	mvmB,
		void *	B,
		uint	rows,
		uint	cols
	)

Approximate the spectral norm $\|A-B\|_2$ of the difference of two matrices $A$ and $B$ .

The spectral norm is approximated by applying a few steps of the power iteration to the matrix $(A-B)^* (A-B)$ and computing the square root of the resulting eigenvalue approximation.

Parameters

mvmA	Callback function for evaluating `A` and its adjoint.
A	Matrix .
mvmB	Callback function for evaluating `B` and its adjoint.
B	Matrix .
rows	Number of rows of either `A` and `B`.
cols	Number of columns of either `A` and `B`.

Returns: Approximation of $\|A-B\|_2$ .

real norm2diff_pre_matrix	(	mvm_t	mvmA,
		void *	A,
		prcd_t	evalB,
		prcd_t	evaltransB,
		void *	B,
		uint	rows,
		uint	cols
	)

Approximate the spectral norm $\|A-B\|_2$ of the difference of two matrices $A$ and $B$ . The matrix $B$ is given as a factorization and can be applied to some vector.

The spectral norm is approximated by applying a few steps of the power iteration to the matrix $(A-B)^* (A-B)$ and computing the square root of the resulting eigenvalue approximation. $B$ is usually given as a LR decomposition or cholesky factorization of $A$ .

Parameters

mvmA	Callback function for evaluating `A` and its adjoint.
A	Matrix .
evalB	Callback function for evaluating `B`.
evaltransB	Callback function for evaluating the adjoint of `B`.
B	Factorized matrix .
rows	Number of rows of either `A` and `B`.
cols	Number of columns of either `A` and `B`.

Returns: Approximation of $\|A-B\|_2$ .

real residualnorm_gmres	(	pcavector	rhat,
		uint	k
	)

Returns norm of current residual vector.

Parameters

rhat	Transformed residual. The absolute value of `rhat[k]` is the Euclidean norm of the residual.
k	Current dimension of the Krylov space.

Returns: Norm of the residual.

real residualnorm_pgmres	(	pcavector	rhat,
		uint	k
	)

Returns norm of current preconditioned residual vector.

Parameters

rhat	Transformed residual. The absolute value of `rhat[k]` is the Euclidean norm of the preconditioned residual.
k	Current dimension of the Krylov space.

Returns: Norm of the residual.

void step_bicg	(	addeval_t	addeval,
		addeval_t	addevaltrans,
		void *	matrix,
		pcavector	b,
		pavector	x,
		pavector	r,
		pavector	rt,
		pavector	p,
		pavector	pt,
		pavector	a,
		pavector	at
	)

One step a biconjugate gradient method.

Parameters

addeval	Callback function name
addevaltrans	Callback function name
matrix	untyped pointer to matrix data
b	Right-hand side.
x	Approximate solution.
r	Residual b-Ax
rt	Adjoint residual
p	Search direction
pt	Adjoint search direction
a	auxiliary vector
at	auxiliary vector

void step_bicgstab	(	addeval_t	addeval,
		void *	matrix,
		pcavector	b,
		pavector	x,
		pavector	r,
		pavector	rt,
		pavector	p,
		pavector	a,
		pavector	as
	)

One step a stabilized biconjugate gradient method.

Parameters

addeval	Callback function name
matrix	untyped pointer to matrix data
b	Right-hand side.
x	Approximate solution.
r	Residual b-Ax
rt	Adjoint residual
p	Search direction
a	auxiliary vector
as	auxiliary vector

void step_cg	(	addeval_t	addeval,
		void *	matrix,
		pcavector	b,
		pavector	x,
		pavector	r,
		pavector	p,
		pavector	a
	)

One step of a standard conjugate gradient method.

Parameters

addeval	Callback function name
matrix	untyped pointer to matrix data (A)
b	Right-hand side.
x	Approximate solution.
r	Residual b-Ax
p	Search direction
a	auxiliary vector

void step_gmres	(	addeval_t	addeval,
		void *	matrix,
		pcavector	b,
		pavector	x,
		pavector	rhat,
		pavector	q,
		uint *	kk,
		pamatrix	qr,
		pavector	tau
	)

One step of the GMRES method.

If *kk+1 >= qr->cols, there is no room for the next Arnoldi basis vector and the function returns immediately. It can be restarted using finish_gmres.

Otherwise a new vector is added to the Arnoldi basis and the matrix $Q_{k+1}^* A Q_k$ is updated.

Remarks: This function currently makes no use of b and does not update x. The current residual can be tracked via rhat[*kk]. Once it is sufficiently small, the improved solution x can be obtained by using finish_gmres.

Parameters

addeval	Callback function representing the matrix .
matrix	Data for the `addeval` callback.
b	Right-hand side vector .
x	Initial guess for the solution , will eventually be replaced by an improved approximation.
rhat	Transformed residual. The absolute value of `rhat[*kk]` is the Euclidean norm of the residual.
q	Next vector of the Krylov basis, constructed by Householder's elementary reflectors.
kk	Pointer to current dimension of the Krylov space.
qr	Representation of the Arnoldi basis $Q_{k+1}$ and the transformed matrix $Q_{k+1}^* A Q_k$ .
tau	Scaling factors of elementary reflectors, provided by qrdecomp_amatrix.

void step_pcg	(	addeval_t	addeval,
		void *	matrix,
		prcd_t	prcd,
		void *	pdata,
		pcavector	b,
		pavector	x,
		pavector	r,
		pavector	q,
		pavector	p,
		pavector	a
	)

One step of a preconditioned conjugate gradient method.

Parameters

addeval	Callback function name
matrix	untyped pointer to matrix data
prcd	Callback function name
pdata	untyped pointer to matrix data
b	Right-hand side.
x	Approximate solution.
r	Residual b-Ax
q	Preconditioned residual
p	Search direction
a	auxiliary vector

void step_pgmres	(	addeval_t	addeval,
		void *	matrix,
		prcd_t	prcd,
		void *	pdata,
		pcavector	b,
		pavector	x,
		pavector	rhat,
		pavector	q,
		uint *	kk,
		pamatrix	qr,
		pavector	tau
	)

One step of the preconditioned GMRES method.

If *kk+1 >= qr->cols, there is no room for the next Arnoldi basis vector and the function returns immediately. It can be restarted using finish_gmres.

Otherwise a new vector is added to the Arnoldi basis and the matrix $Q_{k+1}^* N A Q_k$ is updated.

Remarks: This function currently makes no use of b and does not update x. The current preconditioned residual can be tracked via rhat[*kk]. Once it is sufficiently small, the improved solution x can be obtained by using finish_gmres.

Parameters

addeval	Callback function representing the matrix .
matrix	Data for the `addeval` callback.
prcd	Callback function representing the preconditioner .
pdata	Data for the `prcd` callback.
b	Right-hand side vector .
x	Initial guess for the solution , will eventually be replaced by an improved approximation.
rhat	Transformed residual. The absolute value of `rhat[*kk]` is the Euclidean norm of the preconditioned residual.
q	Next vector of the Krylov basis, constructed by Householder's elementary reflectors.
kk	Pointer to current dimension of the Krylov space.
qr	Representation of the Arnoldi basis $Q_{k+1}$ and the transformed matrix $Q_{k+1}^* A Q_k$ .
tau	Scaling factors of elementary reflectors, provided by qrdecomp_amatrix.

void step_puzawa	(	prcd_t	solve_A11,
		void *	matrix_A11,
		mvm_t	mvm_A21,
		void *	matrix_A21,
		prcd_t	prcd,
		void *	pdata,
		pcavector	b1,
		pcavector	b2,
		pavector	x1,
		pavector	x2,
		pavector	r2,
		pavector	q2,
		pavector	p2,
		pavector	a1,
		pavector	s2
	)

Perform one step of the preconditioned Uzawa iteration.

Parameters

solve_A11	Callback for solving $A_{11} x_1 = b_1$
matrix_A11	Data for `solve_A11` callback
mvm_A21	Callback for evaluating $A_{21} x_1$ or $A_{21}^* x_2$
matrix_A21	Data for `mvm_A21` callback
prcd	Callback function name
pdata	untyped pointer to matrix data
b1	Right-hand side vector
b2	Right-hand side vector
x1	Approximate solution vector
x2	Approximate solution vector
r2	Residual of the Schur complement system, may be useful for a stopping criterion
q2	Auxiliary variable of the same dimension as
p2	Search direction for Schur complement system
a1	Auxiliary variable of the same dimension as
s2	Auxiliary variable of the same dimension as

void step_uzawa	(	prcd_t	solve_A11,
		void *	matrix_A11,
		mvm_t	mvm_A21,
		void *	matrix_A21,
		pcavector	b1,
		pcavector	b2,
		pavector	x1,
		pavector	x2,
		pavector	r2,
		pavector	p2,
		pavector	a1,
		pavector	s2
	)

Perform one step of the standard Uzawa iteration.

Parameters

solve_A11	Callback for solving $A_{11} x_1 = b_1$
matrix_A11	Data for `solve_A11` callback
mvm_A21	Callback for evaluating $A_{21} x_1$ or $A_{21}^* x_2$
matrix_A21	Data for `mvm_A21` callback
b1	Right-hand side vector
b2	Right-hand side vector
x1	Approximate solution vector
x2	Approximate solution vector
r2	Residual of the Schur complement system, may be useful for a stopping criterion
p2	Search direction for Schur complement system
a1	Auxiliary variable of the same dimension as
s2	Auxiliary variable of the same dimension as

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