GALAHAD SBLS package#

purpose#

Given a block, symmetric matrix

\[\begin{split}K_{H} = \begin{pmatrix}H & A^T \\ A & - C\end{pmatrix},\end{split}\]
the sbls package constructs a variety of preconditioners of the form
\[\begin{split}K_{G} = \begin{pmatrix}G & A^T \\ A & - C\end{pmatrix}.\end{split}\]
Here, the leading-block matrix \(G\) is a suitably-chosen approximation to \(H\); it may either be prescribed explicitly, in which case a symmetric indefinite factorization of \(K_{G}\) will be formed using the GALAHAD package SLS, or implicitly, by requiring certain sub-blocks of \(G\) be zero. In the latter case, a factorization of \(K_{G}\) will be obtained implicitly (and more efficiently) without recourse to sls. In particular, for implicit preconditioners, a reordering
\[\begin{split}K_{G} = P \begin{pmatrix} G_{11}^{} & G_{21}^T & A_1^T \\ G_{21}^{} & G_{22}^{} & A_2^T \\ A_{1}^{} & A_{2}^{} & - C \end{pmatrix} P^T\end{split}\]
involving a suitable permutation \(P\) will be found, for some invertible sub-block (“basis”) \(A_1\) of the columns of \(A\); the selection and factorization of \(A_1\) uses the package uls. Once the preconditioner has been constructed, solutions to the preconditioning system
\[\begin{split}\begin{pmatrix}G & A^T \\ A & - C\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} = \begin{pmatrix}a \\ b\end{pmatrix}\end{split}\]
may be obtained by the package. Full advantage is taken of any zero coefficients in the matrices \(H\), \(A\) and \(C\).

See Section 4 of $GALAHAD/doc/sbls.pdf for additional details.

method#

The method used depends on whether an explicit or implicit factorization is required. In the explicit case, the package is really little more than a wrapper for the symmetric, indefinite linear solver SLS in which the system matrix \(K_G\) is assembled from its constituents \(A\), \(C\) and whichever \(G\) is requested by the user. Implicit-factorization preconditioners are more involved, and there is a large variety of different possibilities. The essential ideas are described in detail in

H. S. Dollar, N. I. M. Gould and A. J. Wathen. ``On implicit-factorization constraint preconditioners’’. In Large Scale Nonlinear Optimization (G. Di Pillo and M. Roma, eds.) Springer Series on Nonconvex Optimization and Its Applications, Vol. 83, Springer Verlag (2006) 61–82

and

H. S. Dollar, N. I. M. Gould, W. H. A. Schilders and A. J. Wathen ``On iterative methods and implicit-factorization preconditioners for regularized saddle-point systems’’. SIAM Journal on Matrix Analysis and Applications 28(1) (2006) 170–189.

The range-space factorization is based upon the decomposition

\[\begin{split}K_G = \begin{pmatrix}G & 0 \\ A & I\end{pmatrix} \begin{pmatrix}G^{-1} & 0 \\ 0 & - S\end{pmatrix} \begin{pmatrix}G & A^T \\ 0 & I\end{pmatrix},\end{split}\]
where the ``Schur complement’’ \(S = C + A G^{-1} A^T\). Such a method requires that \(S\) is easily invertible. This is often the case when \(G\) is a diagonal matrix, in which case \(S\) is frequently sparse, or when \(m \ll n\) in which case \(S\) is small and a dense Cholesky factorization may be used.

When \(C = 0\), the null-space factorization is based upon the decomposition

\[\begin{split}K_{G} = P \begin{pmatrix} G_{11}^{} & 0 & I \\ G_{21}^{} & I & A_{2}^{T} A_{1}^{-T} \\ A_{1}^{} & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & I \\ \;\;\; 0 \;\; & \;\; R \;\; & 0 \\ I & 0 & - G_{11}^{} \end{pmatrix} \begin{pmatrix} G_{11}^{} & G_{21}^T & A_{1}^T \\ 0 & I & 0 \\ I & A_{1}^{-1} A_{2}^{} & 0 \end{pmatrix} P^T,\end{split}\]
where the ``reduced Hessian’’
\[\begin{split}R = ( - A_{2}^{T} A_1^{-T} \;\; I ) \begin{pmatrix}G_{11}^{} & G_{21}^{T} \\ G_{21}^{} & G_{22}^{}\end{pmatrix} \begin{pmatrix}- A_1^{-1} A_2^{} \\ I\end{pmatrix}\end{split}\]
and \(P\) is a suitably-chosen permutation for which \(A_1\) is invertible. The method is most useful when \(m \approx n\) as then the dimension of \(R\) is small and a dense Cholesky factorization may be used.

matrix storage#

The unsymmetric \(m\) by \(n\) matrix \(A\) may be presented and stored in a variety of convenient input formats.

Dense storage format: The matrix \(A\) is stored as a compact dense matrix by rows, that is, the values of the entries of each row in turn are stored in order within an appropriate real one-dimensional array. In this case, component \(n \ast i + j\) of the storage array A_val will hold the value \(A_{ij}\) for \(0 \leq i \leq m-1\), \(0 \leq j \leq n-1\). The string A_type = ‘dense’ should be specified.

Dense by columns storage format: The matrix \(A\) is stored as a compact dense matrix by columns, that is, the values of the entries of each column in turn are stored in order within an appropriate real one-dimensional array. In this case, component \(m \ast j + i\) of the storage array A_val will hold the value \(A_{ij}\) for \(0 \leq i \leq m-1\), $0 \leq j \leq n-1 The string A_type = ‘dense_by_columns’ should be specified.

Sparse co-ordinate storage format: Only the nonzero entries of the matrices are stored. For the \(l\)-th entry, \(0 \leq l \leq ne-1\), of \(A\), its row index i, column index j and value \(A_{ij}\), \(0 \leq i \leq m-1\), \(0 \leq j \leq n-1\), are stored as the \(l\)-th components of the integer arrays A_row and A_col and real array A_val, respectively, while the number of nonzeros is recorded as A_ne = \(ne\). The string A_type = ‘coordinate’should be specified.

Sparse row-wise storage format: Again only the nonzero entries are stored, but this time they are ordered so that those in row i appear directly before those in row i+1. For the i-th row of \(A\) the i-th component of the integer array A_ptr holds the position of the first entry in this row, while A_ptr(m) holds the total number of entries. The column indices j, \(0 \leq j \leq n-1\), and values \(A_{ij}\) of the nonzero entries in the i-th row are stored in components l = A_ptr(i), \(\ldots\), A_ptr(i+1)-1, \(0 \leq i \leq m-1\), of the integer array A_col, and real array A_val, respectively. For sparse matrices, this scheme almost always requires less storage than its predecessor. The string A_type = ‘sparse_by_rows’ should be specified.

Sparse column-wise storage format: Once again only the nonzero entries are stored, but this time they are ordered so that those in column j appear directly before those in column j+1. For the j-th column of \(A\) the j-th component of the integer array A_ptr holds the position of the first entry in this column, while A_ptr(n) holds the total number of entries. The row indices i, \(0 \leq i \leq m-1\), and values \(A_{ij}\) of the nonzero entries in the j-th columnsare stored in components l = A_ptr(j), \(\ldots\), A_ptr(j+1)-1, \(0 \leq j \leq n-1\), of the integer array A_row, and real array A_val, respectively. As before, for sparse matrices, this scheme almost always requires less storage than the co-ordinate format. The string A_type = ‘sparse_by_columns’ should be specified.

The symmetric \(n\) by \(n\) matrix \(H\), as well as the \(m\) by \(m\) matrix \(C\), may also be presented and stored in a variety of formats. But crucially symmetry is exploited by only storing values from the lower triangular part (i.e, those entries that lie on or below the leading diagonal). We focus on \(H\), but everything we say applied equally to \(C\).

Dense storage format: The matrix \(H\) is stored as a compact dense matrix by rows, that is, the values of the entries of each row in turn are stored in order within an appropriate real one-dimensional array. Since \(H\) is symmetric, only the lower triangular part (that is the part \(H_{ij}\) for \(0 \leq j \leq i \leq n-1\)) need be held. In this case the lower triangle should be stored by rows, that is component \(i * i / 2 + j\) of the storage array H_val will hold the value \(H_{ij}\) (and, by symmetry, \(H_{ji}\)) for \(0 \leq j \leq i \leq n-1\). The string H_type = ‘dense’ should be specified.

Sparse co-ordinate storage format: Only the nonzero entries of the matrices are stored. For the \(l\)-th entry, \(0 \leq l \leq ne-1\), of \(H\), its row index i, column index j and value \(H_{ij}\), \(0 \leq j \leq i \leq n-1\), are stored as the \(l\)-th components of the integer arrays H_row and H_col and real array H_val, respectively, while the number of nonzeros is recorded as H_ne = \(ne\). Note that only the entries in the lower triangle should be stored. The string H_type = ‘coordinate’ should be specified.

Sparse row-wise storage format: Again only the nonzero entries are stored, but this time they are ordered so that those in row i appear directly before those in row i+1. For the i-th row of \(H\) the i-th component of the integer array H_ptr holds the position of the first entry in this row, while H_ptr(n) holds the total number of entries. The column indices j, \(0 \leq j \leq i\), and values \(H_{ij}\) of the entries in the i-th row are stored in components l = H_ptr(i), …, H_ptr(i+1)-1 of the integer array H_col, and real array H_val, respectively. Note that as before only the entries in the lower triangle should be stored. For sparse matrices, this scheme almost always requires less storage than its predecessor. The string H_type = ‘sparse_by_rows’ should be specified.

Diagonal storage format: If \(H\) is diagonal (i.e., \(H_{ij} = 0\) for all \(0 \leq i \neq j \leq n-1\)) only the diagonals entries \(H_{ii}\), \(0 \leq i \leq n-1\) need be stored, and the first n components of the array H_val may be used for the purpose. The string H_type = ‘diagonal’ should be specified.

Multiples of the identity storage format: If \(H\) is a multiple of the identity matrix, (i.e., \(H = \alpha I\) where \(I\) is the n by n identity matrix and \(\alpha\) is a scalar), it suffices to store \(\alpha\) as the first component of H_val. The string H_type = ‘scaled_identity’ should be specified.

The identity matrix format: If \(H\) is the identity matrix, no values need be stored. The string H_type = ‘identity’ should be specified.

The zero matrix format: The same is true if \(H\) is the zero matrix, but now the string H_type = ‘zero’ or ‘none’ should be specified.

introduction to function calls#

To solve a given problem, functions from the sbls package must be called in the following order:

See the examples section for illustrations of use.

callable functions#

overview of functions provided#

// namespaces

namespace conf;

// typedefs

typedef float spc_;
typedef double rpc_;
typedef int ipc_;

// structs

struct sbls_control_type;
struct sbls_inform_type;
struct sbls_time_type;

// global functions

void sbls_initialize(
    void **data,
    struct sbls_control_type* control,
    ipc_ *status
);

void sbls_read_specfile(
    struct sbls_control_type* control,
    const char specfile[]
);

void sbls_import(
    struct sbls_control_type* control,
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ m,
    const char H_type[],
    ipc_ H_ne,
    const ipc_ H_row[],
    const ipc_ H_col[],
    const ipc_ H_ptr[],
    const char A_type[],
    ipc_ A_ne,
    const ipc_ A_row[],
    const ipc_ A_col[],
    const ipc_ A_ptr[],
    const char C_type[],
    ipc_ C_ne,
    const ipc_ C_row[],
    const ipc_ C_col[],
    const ipc_ C_ptr[]
);

void sbls_reset_control(
    struct sbls_control_type* control,
    void **data,
    ipc_ *status
);

void sbls_factorize_matrix(
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ h_ne,
    const rpc_ H_val[],
    ipc_ a_ne,
    const rpc_ A_val[],
    ipc_ c_ne,
    const rpc_ C_val[],
    const rpc_ D[]
);

void sbls_solve_system(void **data, ipc_ *status, ipc_ n, ipc_ m, rpc_ sol[]);
void sbls_information(void **data, struct sbls_inform_type* inform, ipc_ *status);

void sbls_terminate(
    void **data,
    struct sbls_control_type* control,
    struct sbls_inform_type* inform
);

typedefs#

typedef float spc_

spc_ is real single precision

typedef double rpc_

rpc_ is the real working precision used, but may be changed to float by defining the preprocessor variable SINGLE.

typedef int ipc_

ipc_ is the default integer word length used, but may be changed to int64_t by defining the preprocessor variable INTEGER_64.

function calls#

void sbls_initialize(
    void **data,
    struct sbls_control_type* control,
    ipc_ *status
)

Set default control values and initialize private data

Parameters:

data

holds private internal data

control

is a struct containing control information (see sbls_control_type)

status

is a scalar variable of type ipc_, that gives the exit status from the package. Possible values are (currently):

  • 0

    The initialization was successful.

 
void sbls_read_specfile(
    struct sbls_control_type* control,
    const char specfile[]
)

Read the content of a specification file, and assign values associated with given keywords to the corresponding control parameters. An in-depth discussion of specification files is available, and a detailed list of keywords with associated default values is provided in $GALAHAD/src/sbls/SBLS.template. See also Table 2.1 in the Fortran documentation provided in $GALAHAD/doc/sbls.pdf for a list of how these keywords relate to the components of the control structure.

Parameters:

control

is a struct containing control information (see sbls_control_type)

specfile

is a character string containing the name of the specification file

 
void sbls_import(
    struct sbls_control_type* control,
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ m,
    const char H_type[],
    ipc_ H_ne,
    const ipc_ H_row[],
    const ipc_ H_col[],
    const ipc_ H_ptr[],
    const char A_type[],
    ipc_ A_ne,
    const ipc_ A_row[],
    const ipc_ A_col[],
    const ipc_ A_ptr[],
    const char C_type[],
    ipc_ C_ne,
    const ipc_ C_row[],
    const ipc_ C_col[],
    const ipc_ C_ptr[]
)

Import structural matrix data into internal storage prior to solution.

Parameters:

control

is a struct whose members provide control paramters for the remaining prcedures (see sbls_control_type)

data

holds private internal data

status

is a scalar variable of type ipc_, that gives the exit status from the package. Possible values are:

  • 0

    The import was successful.

  • -1

    An allocation error occurred. A message indicating the offending array is written on unit control.error, and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively.

  • -2

    A deallocation error occurred. A message indicating the offending array is written on unit control.error and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively.

  • -3

    The restrictions n > 0 or m > 0 or requirement that a type contains its relevant string ‘dense’, ‘coordinate’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated.

n

is a scalar variable of type ipc_, that holds the number of rows in the symmetric matrix \(H\).

m

is a scalar variable of type ipc_, that holds the number of rows in the symmetric matrix \(C\).

H_type

is a one-dimensional array of type char that specifies the symmetric storage scheme used for the matrix \(H\). It should be one of ‘coordinate’, ‘sparse_by_rows’, ‘dense’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’, the latter pair if \(H=0\); lower or upper case variants are allowed.

H_ne

is a scalar variable of type ipc_, that holds the number of entries in the lower triangular part of \(H\) in the sparse co-ordinate storage scheme. It need not be set for any of the other schemes.

H_row

is a one-dimensional array of size H_ne and type ipc_, that holds the row indices of the lower triangular part of \(H\) in the sparse co-ordinate storage scheme. It need not be set for any of the other three schemes, and in this case can be NULL.

H_col

is a one-dimensional array of size H_ne and type ipc_, that holds the column indices of the lower triangular part of \(H\) in either the sparse co-ordinate, or the sparse row-wise storage scheme. It need not be set when the dense, diagonal or (scaled) identity storage schemes are used, and in this case can be NULL.

H_ptr

is a one-dimensional array of size n+1 and type ipc_, that holds the starting position of each row of the lower triangular part of \(H\), as well as the total number of entries, in the sparse row-wise storage scheme. It need not be set when the other schemes are used, and in this case can be NULL.

A_type

is a one-dimensional array of type char that specifies the symmetric storage scheme used for the matrix \(A\). It should be one of ‘coordinate’, ‘sparse_by_rows’, ‘dense’ or ‘absent’, the latter if access to the Jacobian is via matrix-vector products; lower or upper case variants are allowed.

A_ne

is a scalar variable of type ipc_, that holds the number of entries in \(A\) in the sparse co-ordinate storage scheme. It need not be set for any of the other schemes.

A_row

is a one-dimensional array of size A_ne and type ipc_, that holds the row indices of \(A\) in the sparse co-ordinate storage scheme. It need not be set for any of the other schemes, and in this case can be NULL.

A_col

is a one-dimensional array of size A_ne and type ipc_, that holds the column indices of \(A\) in either the sparse co-ordinate, or the sparse row-wise storage scheme. It need not be set when the dense or diagonal storage schemes are used, and in this case can be NULL.

A_ptr

is a one-dimensional array of size n+1 and type ipc_, that holds the starting position of each row of \(A\), as well as the total number of entries, in the sparse row-wise storage scheme. It need not be set when the other schemes are used, and in this case can be NULL.

C_type

is a one-dimensional array of type char that specifies the symmetric storage scheme used for the matrix \(C\). It should be one of ‘coordinate’, ‘sparse_by_rows’, ‘dense’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’, the latter pair if \(C=0\); lower or upper case variants are allowed.

C_ne

is a scalar variable of type ipc_, that holds the number of entries in the lower triangular part of \(C\) in the sparse co-ordinate storage scheme. It need not be set for any of the other schemes.

C_row

is a one-dimensional array of size C_ne and type ipc_, that holds the row indices of the lower triangular part of \(C\) in the sparse co-ordinate storage scheme. It need not be set for any of the other three schemes, and in this case can be NULL.

C_col

is a one-dimensional array of size C_ne and type ipc_, that holds the column indices of the lower triangular part of \(C\) in either the sparse co-ordinate, or the sparse row-wise storage scheme. It need not be set when the dense, diagonal or (scaled) identity storage schemes are used, and in this case can be NULL.

C_ptr

is a one-dimensional array of size n+1 and type ipc_, that holds the starting position of each row of the lower triangular part of \(C\), as well as the total number of entries, in the sparse row-wise storage scheme. It need not be set when the other schemes are used, and in this case can be NULL.

 
void sbls_reset_control(
    struct sbls_control_type* control,
    void **data,
    ipc_ *status
)

Reset control parameters after import if required.

Parameters:

control

is a struct whose members provide control paramters for the remaining prcedures (see sbls_control_type)

data

holds private internal data

status

is a scalar variable of type ipc_, that gives the exit status from the package. Possible values are:

  • 0

    The import was successful.

 
void sbls_factorize_matrix(
    void **data,
    ipc_ *status,
    ipc_ n,
    ipc_ h_ne,
    const rpc_ H_val[],
    ipc_ a_ne,
    const rpc_ A_val[],
    ipc_ c_ne,
    const rpc_ C_val[],
    const rpc_ D[]
)

Form and factorize the block matrix

\[\begin{split}K_{G} = \begin{pmatrix}G & A^T \\ A & - C\end{pmatrix}\end{split}\]
for some appropriate matrix \(G\).

Parameters:

data

holds private internal data

status

is a scalar variable of type ipc_, that gives the exit status from the package.

Possible values are:

  • 0

    The factors were generated successfully.

  • -1

    An allocation error occurred. A message indicating the offending array is written on unit control.error, and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively.

  • -2

    A deallocation error occurred. A message indicating the offending array is written on unit control.error and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively.

  • -3

    The restrictions n > 0 and m > 0 or requirement that a type contains its relevant string ‘dense’, ‘coordinate’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated.

  • -9

    An error was reported by SLS analyse. The return status from SLS analyse is given in inform.sls_inform.status. See the documentation for the GALAHAD package SLS for further details.

  • -10

    An error was reported by SLS_factorize. The return status from SLS factorize is given in inform.sls_inform.status. See the documentation for the GALAHAD package SLS for further details.

  • -13

    An error was reported by ULS_factorize. The return status from ULS_factorize is given in inform.uls_factorize_status. See the documentation for the GALAHAD package ULS for further details.

  • -15

    The computed preconditioner \(K_G\) is singular and is thus unsuitable.

  • -20

    The computed preconditioner \(K_G\) has the wrong inertia and is thus unsuitable.

  • -24

    An error was reported by the GALAHAD package SORT_reorder_by_rows. The return status from SORT_reorder_by_rows is given in inform.sort_status. See the documentation for the GALAHAD package SORT for further details.

n

is a scalar variable of type ipc_, that holds the number of rows in the symmetric matrix \(H\).

h_ne

is a scalar variable of type ipc_, that holds the number of entries in the lower triangular part of the symmetric matrix \(H\).

H_val

is a one-dimensional array of size h_ne and type rpc_, that holds the values of the entries of the lower triangular part of the symmetric matrix \(H\) in any of the available storage schemes

a_ne

is a scalar variable of type ipc_, that holds the number of entries in the unsymmetric matrix \(A\).

A_val

is a one-dimensional array of size a_ne and type rpc_, that holds the values of the entries of the unsymmetric matrix \(A\) in any of the available storage schemes.

c_ne

is a scalar variable of type ipc_, that holds the number of entries in the lower triangular part of the symmetric matrix \(C\).

C_val

is a one-dimensional array of size c_ne and type rpc_, that holds the values of the entries of the lower triangular part of the symmetric matrix \(C\) in any of the available storage schemes

D

is a one-dimensional array of size n and type rpc_, that holds the values of the entries of the diagonal matrix \(D\) that is required if the user has specified control.preconditioner = 5. It need not be set otherwise.

 
void sbls_solve_system(void **data, ipc_ *status, ipc_ n, ipc_ m, rpc_ sol[])

Solve the block linear system

\[\begin{split}\begin{pmatrix}G & A^T \\ A & - C\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} = \begin{pmatrix}a \\ b\end{pmatrix}.\end{split}\]

Parameters:

data

holds private internal data

status

is a scalar variable of type ipc_, that gives the exit status from the package.

Possible values are:

  • 0

    The required solution was obtained.

  • -1

    An allocation error occurred. A message indicating the offending array is written on unit control.error, and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively.

  • -2

    A deallocation error occurred. A message indicating the offending array is written on unit control.error and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively.

  • -11

    An error was reported by SLS_solve. The return status from SLS solve is given in inform.sls_inform.status. See the documentation for the GALAHAD package SLS for further details.

  • -14

    An error was reported by ULS_solve. The return status from ULS_solve is given in inform.uls_solve_status. See the documentation for the GALAHAD package ULS for further details.

n

is a scalar variable of type ipc_, that holds the number of entries in the vector \(a\).

m

is a scalar variable of type ipc_, that holds the number of entries in the vector \(b\).

sol

is a one-dimensional array of size n + m and type double. on entry, its first n entries must hold the vector \(a\), and the following entries must hold the vector \(b\). On a successful exit, its first n entries contain the solution components \(x\), and the following entries contain the components \(y\).

 
void sbls_information(void **data, struct sbls_inform_type* inform, ipc_ *status)

Provides output information

Parameters:

data

holds private internal data

inform

is a struct containing output information (see sbls_inform_type)

status

is a scalar variable of type ipc_, that gives the exit status from the package. Possible values are (currently):

  • 0

    The values were recorded successfully

 
void sbls_terminate(
    void **data,
    struct sbls_control_type* control,
    struct sbls_inform_type* inform
)

Deallocate all internal private storage

Parameters:

data

holds private internal data

control

is a struct containing control information (see sbls_control_type)

inform

is a struct containing output information (see sbls_inform_type)

available structures#

sbls_control_type structure#

#include <galahad_sbls.h>

struct sbls_control_type {
    // fields

    bool f_indexing;
    ipc_ error;
    ipc_ out;
    ipc_ print_level;
    ipc_ indmin;
    ipc_ valmin;
    ipc_ len_ulsmin;
    ipc_ itref_max;
    ipc_ maxit_pcg;
    ipc_ new_a;
    ipc_ new_h;
    ipc_ new_c;
    ipc_ preconditioner;
    ipc_ semi_bandwidth;
    ipc_ factorization;
    ipc_ max_col;
    ipc_ scaling;
    ipc_ ordering;
    rpc_ pivot_tol;
    rpc_ pivot_tol_for_basis;
    rpc_ zero_pivot;
    rpc_ static_tolerance;
    rpc_ static_level;
    rpc_ min_diagonal;
    rpc_ stop_absolute;
    rpc_ stop_relative;
    bool remove_dependencies;
    bool find_basis_by_transpose;
    bool affine;
    bool allow_singular;
    bool perturb_to_make_definite;
    bool get_norm_residual;
    bool check_basis;
    bool space_critical;
    bool deallocate_error_fatal;
    char symmetric_linear_solver[31];
    char definite_linear_solver[31];
    char unsymmetric_linear_solver[31];
    char prefix[31];
    struct sls_control_type sls_control;
    struct uls_control_type uls_control;
};

detailed documentation#

control derived type as a C struct

components#

bool f_indexing

use C or Fortran sparse matrix indexing

ipc_ error

unit for error messages

ipc_ out

unit for monitor output

ipc_ print_level

controls level of diagnostic output

ipc_ indmin

initial estimate of integer workspace for SLS (obsolete)

ipc_ valmin

initial estimate of real workspace for SLS (obsolete)

ipc_ len_ulsmin

initial estimate of workspace for ULS (obsolete)

ipc_ itref_max

maximum number of iterative refinements with preconditioner allowed

ipc_ maxit_pcg

maximum number of projected CG iterations allowed

ipc_ new_a

how much has \(A\) changed since last factorization: 0 = not changed, 1 = values changed, 2 = structure changed

ipc_ new_h

how much has \(H\) changed since last factorization: 0 = not changed, 1 = values changed, 2 = structure changed

ipc_ new_c

how much has \(C\) changed since last factorization: 0 = not changed, 1 = values changed, 2 = structure changed

ipc_ preconditioner

which preconditioner to use:

  • 0 selected automatically

  • 1 explicit with \(G = I\)

  • 2 explicit with \(G = H\)

  • 3 explicit with \(G =\) diag(max(\(H\),min_diag))

  • 4 explicit with \(G =\) band \((H)\)

  • 5 explicit with \(G =\) (optional, diagonal) \(D\)

  • 11 explicit with \(G_{11} = 0\), \(G_{21} = 0\), \(G_{22} = H_{22}\)

  • 12 explicit with \(G_{11} = 0\), \(G_{21} = H_{21}\), \(G_{22} = H_{22}\)

  • -1 implicit with \(G_{11} = 0\), \(G_{21} = 0\), \(G_{22} = I\)

  • -2 implicit with \(G_{11} = 0\), \(G_{21} = 0\), \(G_{22} = H_{22}\)

ipc_ semi_bandwidth

the semi-bandwidth for band(H)

ipc_ factorization

the explicit factorization used:

  • 0 selected automatically

  • 1 Schur-complement if \(G\) is diagonal and successful otherwise augmented system

  • 2 augmented system

  • 3 null-space

  • 4 Schur-complement if \(G\) is diagonal and successful otherwise failure

  • 5 Schur-complement with pivoting if \(G\) is diagonal and successful otherwise failure

ipc_ max_col

maximum number of nonzeros in a column of \(A\) for Schur-complement factorization

ipc_ scaling

not used at present

ipc_ ordering

see scaling

rpc_ pivot_tol

the relative pivot tolerance used by ULS (obsolete)

rpc_ pivot_tol_for_basis

the relative pivot tolerance used by ULS when determining the basis matrix

rpc_ zero_pivot

the absolute pivot tolerance used by ULS (obsolete)

rpc_ static_tolerance

not used at present

rpc_ static_level

see static_tolerance

rpc_ min_diagonal

the minimum permitted diagonal in diag(max(\(H\),min_diag))

rpc_ stop_absolute

the required absolute and relative accuracies

rpc_ stop_relative

see stop_absolute

bool remove_dependencies

preprocess equality constraints to remove linear dependencies

bool find_basis_by_transpose

determine implicit factorization preconditioners using a basis of A found by examining A’s transpose

bool affine

can the right-hand side \(c\) be assumed to be zero?

bool allow_singular

do we tolerate “singular” preconditioners?

bool perturb_to_make_definite

if the initial attempt at finding a preconditioner is unsuccessful, should the diagonal be perturbed so that a second attempt succeeds?

bool get_norm_residual

compute the residual when applying the preconditioner?

bool check_basis

if an implicit or null-space preconditioner is used, assess and correct for ill conditioned basis matrices

bool space_critical

if space is critical, ensure allocated arrays are no bigger than needed

bool deallocate_error_fatal

exit if any deallocation fails

char symmetric_linear_solver[31]

the name of the symmetric-indefinite linear equation solver used. Possible choices are currently: ‘sils’, ‘ma27’, ‘ma57’, ‘ma77’, ‘ma86’, ‘ma97’, ‘ssids’, ‘mumps’, ‘pardiso’, ‘mkl_pardiso’, ‘pastix’, ‘wsmp’, and ‘sytr’, although only ‘sytr’ and, for OMP 4.0-compliant compilers, ‘ssids’ are installed by default; others are easily installed (see README.external). More details of the capabilities of each solver are provided in the documentation for galahad_sls.

char definite_linear_solver[31]

the name of the definite linear equation solver used. Possible choices are currently: ‘sils’, ‘ma27’, ‘ma57’, ‘ma77’, ‘ma86’, ‘ma87’, ‘ma97’, ‘ssids’, ‘mumps’, ‘pardiso’, ‘mkl_pardiso’, ‘pastix’, ‘wsmp’, ‘potr’, ‘sytr’ and ‘pbtr’, although only ‘potr’, ‘sytr’, ‘pbtr’ and, for OMP 4.0-compliant compilers, ‘ssids’ are installed by default; others are easily installed (see README.external). More details of the capabilities of each solver are provided in the documentation for galahad_sls.

char unsymmetric_linear_solver[31]

the name of the unsymmetric linear equation solver used. Possible choices are currently: ‘gls’, ‘ma48’ and ‘getr’, although only ‘getr’ is installed by default; others are easily installed (see README.external). More details of the capabilities of each solver are provided in the documentation for galahad_uls.

char prefix[31]

all output lines will be prefixed by prefix(2:LEN(TRIM(.prefix))-1) where prefix contains the required string enclosed in quotes, e.g. “string” or ‘string’

struct sls_control_type sls_control

control parameters for SLS

struct uls_control_type uls_control

control parameters for ULS

sbls_time_type structure#

#include <galahad_sbls.h>

struct sbls_time_type {
    // fields

    rpc_ total;
    rpc_ form;
    rpc_ factorize;
    rpc_ apply;
    rpc_ clock_total;
    rpc_ clock_form;
    rpc_ clock_factorize;
    rpc_ clock_apply;
};

detailed documentation#

time derived type as a C struct

components#

rpc_ total

total cpu time spent in the package

rpc_ form

cpu time spent forming the preconditioner \(K_G\)

rpc_ factorize

cpu time spent factorizing \(K_G\)

rpc_ apply

cpu time spent solving linear systems inolving \(K_G\)

rpc_ clock_total

total clock time spent in the package

rpc_ clock_form

clock time spent forming the preconditioner \(K_G\)

rpc_ clock_factorize

clock time spent factorizing \(K_G\)

rpc_ clock_apply

clock time spent solving linear systems inolving \(K_G\)

sbls_inform_type structure#

#include <galahad_sbls.h>

struct sbls_inform_type {
    // fields

    ipc_ status;
    ipc_ alloc_status;
    char bad_alloc[81];
    ipc_ sort_status;
    int64_t factorization_integer;
    int64_t factorization_real;
    ipc_ preconditioner;
    ipc_ factorization;
    ipc_ d_plus;
    ipc_ rank;
    bool rank_def;
    bool perturbed;
    ipc_ iter_pcg;
    rpc_ norm_residual;
    bool alternative;
    struct sbls_time_type time;
    struct sls_inform_type sls_inform;
    struct uls_inform_type uls_inform;
};

detailed documentation#

inform derived type as a C struct

components#

ipc_ status

return status. See SBLS_form_and_factorize for details

ipc_ alloc_status

the status of the last attempted allocation/deallocation

char bad_alloc[81]

the name of the array for which an allocation/deallocation error occurred

ipc_ sort_status

the return status from the sorting routines

int64_t factorization_integer

the total integer workspace required for the factorization

int64_t factorization_real

the total real workspace required for the factorization

ipc_ preconditioner

the preconditioner used

ipc_ factorization

the factorization used

ipc_ d_plus

how many of the diagonals in the factorization are positive

ipc_ rank

the computed rank of \(A\)

bool rank_def

is the matrix A rank defficient?

bool perturbed

has the used preconditioner been perturbed to guarantee correct inertia?

ipc_ iter_pcg

the total number of projected CG iterations required

rpc_ norm_residual

the norm of the residual

bool alternative

has an “alternative” \(y\) : \(K y = 0\) and \(y^T c > 0\) been found when trying to solve \(K y = c\) for generic \(K\)?

struct sbls_time_type time

timings (see above)

struct sls_inform_type sls_inform

inform parameters from the GALAHAD package SLS used

struct uls_inform_type uls_inform

inform parameters from the GALAHAD package ULS used

Index

example calls#

This is an example of how to use the package to solve a block system of linear equations; the code is available in $GALAHAD/src/sbls/C/sblst.c . A variety of supported matrix storage formats are shown.

Notice that C-style indexing is used, and that this is flagged by setting control.f_indexing to false. The floating-point type rpc_ is set in galahad_precision.h to double by default, but to float if the preprocessor variable SINGLE is defined. Similarly, the integer type ipc_ from galahad_precision.h is set to int by default, but to int64_t if the preprocessor variable INTEGER_64 is defined.

/* sblst.c */
/* Full test for the SBLS C interface using C sparse matrix indexing */

#include <stdio.h>
#include <math.h>
#include <string.h>
#include "galahad_precision.h"
#include "galahad_cfunctions.h"
#include "galahad_sbls.h"

int main(void) {

    // Derived types
    void *data;
    struct sbls_control_type control;
    struct sbls_inform_type inform;

    // Set problem data
    ipc_ n = 3; // dimension of H
    ipc_ m = 2; // dimension of C
    ipc_ H_ne = 4; // number of elements of H
    ipc_ A_ne = 3; // number of elements of A
    ipc_ C_ne = 3; // number of elements of C
    ipc_ H_dense_ne = 6; // number of elements of H
    ipc_ A_dense_ne = 6; // number of elements of A
    ipc_ C_dense_ne = 3; // number of elements of C
    ipc_ H_row[] = {0, 1, 2, 2}; // row indices, NB lower triangle
    ipc_ H_col[] = {0, 1, 2, 0};
    ipc_ H_ptr[] = {0, 1, 2, 4};
    ipc_ A_row[] = {0, 0, 1};
    ipc_ A_col[] = {0, 1, 2};
    ipc_ A_ptr[] = {0, 2, 3};
    ipc_ C_row[] = {0, 1, 1}; // row indices, NB lower triangle
    ipc_ C_col[] = {0, 0, 1};
    ipc_ C_ptr[] = {0, 1, 3};
    rpc_ H_val[] = {1.0, 2.0, 3.0, 1.0};
    rpc_ A_val[] = {2.0, 1.0, 1.0};
    rpc_ C_val[] = {4.0, 1.0, 2.0};
    rpc_ H_dense[] = {1.0, 0.0, 2.0, 1.0, 0.0, 3.0};
    rpc_ A_dense[] = {2.0, 1.0, 0.0, 0.0, 0.0, 1.0};
    rpc_ C_dense[] = {4.0, 1.0, 2.0};
    rpc_ H_diag[] = {1.0, 1.0, 2.0};
    rpc_ C_diag[] = {4.0, 2.0};
    rpc_ H_scid[] = {2.0};
    rpc_ C_scid[] = {2.0};

    char st = ' ';
    ipc_ status;

    printf(" C sparse matrix indexing\n\n");

    printf(" basic tests of storage formats\n\n");

    for( ipc_ d=1; d <= 7; d++){

        // Initialize SBLS
        sbls_initialize( &data, &control, &status );
        control.preconditioner = 2;
        control.factorization = 2;
        control.get_norm_residual = true;
        strcpy(control.symmetric_linear_solver, "sytr ") ;
        strcpy(control.definite_linear_solver, "sytr ") ;

        // Set user-defined control options
        control.f_indexing = false; // C sparse matrix indexing

        switch(d){
            case 1: // sparse co-ordinate storage
                st = 'C';
                sbls_import( &control, &data, &status, n, m,
                           "coordinate", H_ne, H_row, H_col, NULL,
                           "coordinate", A_ne, A_row, A_col, NULL,
                           "coordinate", C_ne, C_row, C_col, NULL );
                sbls_factorize_matrix( &data, &status, n,
                                       H_ne, H_val,
                                       A_ne, A_val,
                                       C_ne, C_val, NULL );
                break;
            printf(" case %1" i_ipc_ " break\n",d);
            case 2: // sparse by rows
                st = 'R';
                sbls_import( &control, &data, &status, n, m,
                            "sparse_by_rows", H_ne, NULL, H_col, H_ptr,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr,
                            "sparse_by_rows", C_ne, NULL, C_col, C_ptr );
                sbls_factorize_matrix( &data, &status, n,
                                       H_ne, H_val,
                                       A_ne, A_val,
                                       C_ne, C_val, NULL );
                break;
            case 3: // dense
                st = 'D';
                sbls_import( &control, &data, &status, n, m,
                            "dense", H_ne, NULL, NULL, NULL,
                            "dense", A_ne, NULL, NULL, NULL,
                            "dense", C_ne, NULL, NULL, NULL );
                sbls_factorize_matrix( &data, &status, n,
                                       H_dense_ne, H_dense,
                                       A_dense_ne, A_dense,
                                       C_dense_ne, C_dense,
                                       NULL );
                break;
            case 4: // diagonal
                st = 'L';
                sbls_import( &control, &data, &status, n, m,
                            "diagonal", H_ne, NULL, NULL, NULL,
                            "dense", A_ne, NULL, NULL, NULL,
                            "diagonal", C_ne, NULL, NULL, NULL );
                sbls_factorize_matrix( &data, &status, n,
                                       n, H_diag,
                                       A_dense_ne, A_dense,
                                       m, C_diag,
                                       NULL );
                break;

            case 5: // scaled identity
                st = 'S';
                sbls_import( &control, &data, &status, n, m,
                            "scaled_identity", H_ne, NULL, NULL, NULL,
                            "dense", A_ne, NULL, NULL, NULL,
                            "scaled_identity", C_ne, NULL, NULL, NULL );
                sbls_factorize_matrix( &data, &status, n,
                                       1, H_scid,
                                       A_dense_ne, A_dense,
                                       1, C_scid,
                                       NULL );
                break;
            case 6: // identity
                st = 'I';
                sbls_import( &control, &data, &status, n, m,
                            "identity", H_ne, NULL, NULL, NULL,
                            "dense", A_ne, NULL, NULL, NULL,
                            "identity", C_ne, NULL, NULL, NULL );
                sbls_factorize_matrix( &data, &status, n,
                                       0, H_val,
                                       A_dense_ne, A_dense,
                                       0, C_val, NULL );
                break;
            case 7: // zero
                st = 'Z';
                sbls_import( &control, &data, &status, n, m,
                            "identity", H_ne, NULL, NULL, NULL,
                            "dense", A_ne, NULL, NULL, NULL,
                            "zero", C_ne, NULL, NULL, NULL );
                sbls_factorize_matrix( &data, &status, n,
                                       0, H_val,
                                       A_dense_ne, A_dense,
                                       0, NULL, NULL );
                break;
            }

        // Set right-hand side ( a, b )
        rpc_ sol[] = {3.0, 2.0, 4.0, 2.0, 0.0};   // values

        sbls_solve_system( &data, &status, n, m, sol );

        sbls_information( &data, &inform, &status );

        if(inform.status == 0){
            printf("%c: residual = %9.1e status = %1" i_ipc_ "\n",
                   st, inform.norm_residual, inform.status);
        }else{
            printf("%c: SBLS_solve exit status = %1" i_ipc_ "\n", st, inform.status);
        }
        //printf("sol: ");
        //for( ipc_ i = 0; i < n+m; i++) printf("%f ", x[i]);

        // Delete internal workspace
        sbls_terminate( &data, &control, &inform );
    }
}

This is the same example, but now fortran-style indexing is used; the code is available in $GALAHAD/src/sbls/C/sblstf.c .

/* sblstf.c */
/* Full test for the SBLS C interface using Fortran sparse matrix indexing */

#include <stdio.h>
#include <math.h>
#include <string.h>
#include "galahad_precision.h"
#include "galahad_cfunctions.h"
#include "galahad_sbls.h"

int main(void) {

    // Derived types
    void *data;
    struct sbls_control_type control;
    struct sbls_inform_type inform;

    // Set problem data
    ipc_ n = 3; // dimension of H
    ipc_ m = 2; // dimension of C
    ipc_ H_ne = 4; // number of elements of H
    ipc_ A_ne = 3; // number of elements of A
    ipc_ C_ne = 3; // number of elements of C
    ipc_ H_dense_ne = 6; // number of elements of H
    ipc_ A_dense_ne = 6; // number of elements of A
    ipc_ C_dense_ne = 3; // number of elements of C
    ipc_ H_row[] = {1, 2, 3, 3}; // row indices, NB lower triangle
    ipc_ H_col[] = {1, 2, 3, 1};
    ipc_ H_ptr[] = {1, 2, 3, 5};
    ipc_ A_row[] = {1, 1, 2};
    ipc_ A_col[] = {1, 2, 3};
    ipc_ A_ptr[] = {1, 3, 4};
    ipc_ C_row[] = {1, 2, 2}; // row indices, NB lower triangle
    ipc_ C_col[] = {1, 1, 2};
    ipc_ C_ptr[] = {1, 2, 4};
    rpc_ H_val[] = {1.0, 2.0, 3.0, 1.0};
    rpc_ A_val[] = {2.0, 1.0, 1.0};
    rpc_ C_val[] = {4.0, 1.0, 2.0};
    rpc_ H_dense[] = {1.0, 0.0, 2.0, 1.0, 0.0, 3.0};
    rpc_ A_dense[] = {2.0, 1.0, 0.0, 0.0, 0.0, 1.0};
    rpc_ C_dense[] = {4.0, 1.0, 2.0};
    rpc_ H_diag[] = {1.0, 1.0, 2.0};
    rpc_ C_diag[] = {4.0, 2.0};
    rpc_ H_scid[] = {2.0};
    rpc_ C_scid[] = {2.0};

    char st = ' ';
    ipc_ status;

    printf(" Fortran sparse matrix indexing\n\n");

    printf(" basic tests of storage formats\n\n");

    for( ipc_ d=1; d <= 7; d++){

        // Initialize SBLS
        sbls_initialize( &data, &control, &status );
        control.preconditioner = 2;
        control.factorization = 2;
        control.get_norm_residual = true;
        strcpy(control.symmetric_linear_solver, "sytr ") ;
        strcpy(control.definite_linear_solver, "sytr ") ;

        // Set user-defined control options
        control.f_indexing = true; // fortran sparse matrix indexing

        switch(d){
            case 1: // sparse co-ordinate storage
                st = 'C';
                sbls_import( &control, &data, &status, n, m,
                           "coordinate", H_ne, H_row, H_col, NULL,
                           "coordinate", A_ne, A_row, A_col, NULL,
                           "coordinate", C_ne, C_row, C_col, NULL );
                sbls_factorize_matrix( &data, &status, n,
                                       H_ne, H_val,
                                       A_ne, A_val,
                                       C_ne, C_val, NULL );
                break;
            printf(" case %1" i_ipc_ " break\n",d);
            case 2: // sparse by rows
                st = 'R';
                sbls_import( &control, &data, &status, n, m,
                            "sparse_by_rows", H_ne, NULL, H_col, H_ptr,
                            "sparse_by_rows", A_ne, NULL, A_col, A_ptr,
                            "sparse_by_rows", C_ne, NULL, C_col, C_ptr );
                sbls_factorize_matrix( &data, &status, n,
                                       H_ne, H_val,
                                       A_ne, A_val,
                                       C_ne, C_val, NULL );
                break;
            case 3: // dense
                st = 'D';
                sbls_import( &control, &data, &status, n, m,
                            "dense", H_ne, NULL, NULL, NULL,
                            "dense", A_ne, NULL, NULL, NULL,
                            "dense", C_ne, NULL, NULL, NULL );
                sbls_factorize_matrix( &data, &status, n,
                                       H_dense_ne, H_dense,
                                       A_dense_ne, A_dense,
                                       C_dense_ne, C_dense,
                                       NULL );
                break;
            case 4: // diagonal
                st = 'L';
                sbls_import( &control, &data, &status, n, m,
                            "diagonal", H_ne, NULL, NULL, NULL,
                            "dense", A_ne, NULL, NULL, NULL,
                            "diagonal", C_ne, NULL, NULL, NULL );
                sbls_factorize_matrix( &data, &status, n,
                                       n, H_diag,
                                       A_dense_ne, A_dense,
                                       m, C_diag,
                                       NULL );
                break;

            case 5: // scaled identity
                st = 'S';
                sbls_import( &control, &data, &status, n, m,
                            "scaled_identity", H_ne, NULL, NULL, NULL,
                            "dense", A_ne, NULL, NULL, NULL,
                            "scaled_identity", C_ne, NULL, NULL, NULL );
                sbls_factorize_matrix( &data, &status, n,
                                       1, H_scid,
                                       A_dense_ne, A_dense,
                                       1, C_scid,
                                       NULL );
                break;
            case 6: // identity
                st = 'I';
                sbls_import( &control, &data, &status, n, m,
                            "identity", H_ne, NULL, NULL, NULL,
                            "dense", A_ne, NULL, NULL, NULL,
                            "identity", C_ne, NULL, NULL, NULL );
                sbls_factorize_matrix( &data, &status, n,
                                       0, H_val,
                                       A_dense_ne, A_dense,
                                       0, C_val, NULL );
                break;
            case 7: // zero
                st = 'Z';
                sbls_import( &control, &data, &status, n, m,
                            "identity", H_ne, NULL, NULL, NULL,
                            "dense", A_ne, NULL, NULL, NULL,
                            "zero", C_ne, NULL, NULL, NULL );
                sbls_factorize_matrix( &data, &status, n,
                                       0, H_val,
                                       A_dense_ne, A_dense,
                                       0, NULL, NULL );
                break;
            }

        // Set right-hand side ( a, b )
        rpc_ sol[] = {3.0, 2.0, 4.0, 2.0, 0.0};   // values

        sbls_solve_system( &data, &status, n, m, sol );

        sbls_information( &data, &inform, &status );

        if(inform.status == 0){
            printf("%c: residual = %9.1e status = %1" i_ipc_ "\n",
                   st, inform.norm_residual, inform.status);
        }else{
            printf("%c: SBLS_solve exit status = %1" i_ipc_ "\n", st, inform.status);
        }
        //printf("sol: ");
        //for( ipc_ i = 0; i < n+m; i++) printf("%f ", x[i]);

        // Delete internal workspace
        sbls_terminate( &data, &control, &inform );
    }
}