GALAHAD SHA package#
purpose#
The sha package
finds a component-wise secant approximation to the Hessian matrix \(H(x)\),
for which \((H(x))_{i,j} = \partial f^2 (x) / \partial x_i \partial x_j\),
\(1 \leq i, j \leq n\),
using values of the gradient \(g(x) = \nabla_x f(x)\)
of the function \(f(x)\) of \(n\) unknowns \(x = (x_1, \ldots, x_n)^T\)
at a sequence of given distinct \(\{x^{(k)}\}\), \(k \geq 0\).
More specifically, given differences
the package aims to find an approximation \(B\) to \(H(x)\) for which the secant conditions \(B s^{(k)} \approx y^{(k)}\) hold for a chosen set of values \(k\). The methods provided take advantage of the entries in the Hessian that are known to be zero.
The package is particularly intended to allow gradient-based optimization methods, that generate iterates \(x^{(k+1)} = x^{(k)} + s^{(k)}\) based upon the values \(g( x^{(k)})\) for \(k \geq 0\), to build a suitable approximation to the Hessian \(H(x^{(k+1)})\). This then gives the method an opportunity to accelerate the iteration using the Hessian approximation.
See Section 4 of $GALAHAD/doc/sha.pdf for additional details.
method#
The package computes the entries in the each row of \(B\) one at a time. The entries \(b_{ij}\) in row \(i\) may be chosen to
where \({\cal I}_i\) is ideally chosen to be sufficiently large so that (1) has a unique minimizer. Since this requires that there are at least as many \((s^{(k)}, y^{(k)})\) pairs as the maximum number of nonzeros in any row, this may be prohibitive in some cases. We might then be content with a minimum-norm (under-determined) least-squares solution; each row may then be processed in parallel. Or, we may take advantage of the symmetry of the Hessian, and note that if we have already found the values in row \(j\), then the value \(b_{i,j} = b_{j,i}\) in (1) is known before we process row \(i\). Thus by ordering the rows and exploiting symmetry we may reduce the numbers of unknowns in future unprocessed rows.
In the analysis phase, we order the rows by constructing the connectivity graph—a graph comprising nodes \(1\) to \(n\) and edges connecting nodes \(i\) and \(j\) if \(h_{i,j}\) is everywhere nonzero—of \(H(x)\). The nodes are ordered by increasing degree (that is, the number of edges emanating from the node) using a bucket sort. The row chosen to be ordered next corresponds to a node of minimum degree, the node is removed from the graph, the degrees updated efficiently, and the process repeated until all rows have been ordered. This often leads to a significant reduction in the numbers of unknown values in each row as it is processed in turn, but numerical rounding can lead to inaccurate values in some cases. A useful remedy is to process all rows for which there are sufficient \((s^{(k)}, y^{(k)})\) as before, and then process the remaining rows taking into account the symmetry. That is, the rows and columns are rearranged so that the matrix is in block form
the \(( B_{11} \;\; B_{12})\) rows are processed without regard for symmetry but give the \(2,1\) block \(B^T_{12}\), and finally the \(2,2\) block \(B_{22}\) is processed knowing \(B^T_{12}\) again without respecting symmetry. The rows in blocks \(( B_{11} \;\; B_{12})\) and \(B_{22}\) may be computed in parallel. It is also possible to generalise this so that \(B\) is decomposed into \(r\) blocks, and the blocks processed one at a time recursively using the symmetry from previos rows. More details of the precise algorithms (Algorithms 2.1–2.5) are given in the reference below. The linear least-squares problems (1) themselves are solved by a choice of LAPACK packages.
references#
The method is described in detail in
J. M. Fowkes, N. I. M. Gould and J. A. Scott, “Approximating large-scale Hessians using secant equations”. Preprint TR-2024-001, Rutherford Appleton Laboratory.
introduction to function calls#
To solve a given problem, functions from the sha package must be called in the following order:
sha_initialize - provide default control parameters and set up initial data structures
sha_read_specfile (optional) - override control values by reading replacement values from a file
sha_analyse_matrix - set up problem data structures and generate information that will be used when estimating Hessian values
sha_recover_matrix - recover the Hessian approximation
sha_information (optional) - recover information about the solution and solution process
sha_terminate - deallocate data structures
See the examples section for illustrations of use.
callable functions#
overview of functions provided#
// typedefs typedef float spc_; typedef double rpc_; typedef int ipc_; // structs struct sha_control_type; struct sha_inform_type; // function calls void sha_initialize(void **data, struct sha_control_type* control, ipc_ *status); void sha_read_specfile(struct sha_control_type* control, const char specfile[]); void sha_analyse_matrix( struct sha_control_type* control, void **data, ipc_ *status, ipc_ n, ipc_ ne, const ipc_ row[], const ipc_ col[], ipc_ *m ); void sha_recover_matrix( void **data, ipc_ *status, ipc_ ne, ipc_ m, ipc_ ls1, ipc_ ls2, const real_wp_ strans[][ls2], ipc_ ly1, ipc_ ly2, const real_wp_ ytrans[][ly2], real_wp_ val[], const ipc_ order[] ); void sha_information(void **data, struct sha_inform_type* inform, ipc_ *status); void sha_terminate( void **data, struct sha_control_type* control, struct sha_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 sha_initialize(void **data, struct sha_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 sha_control_type) |
status |
is a scalar variable of type ipc_, that gives the exit status from the package. Possible values are (currently):
|
void sha_read_specfile(struct sha_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/xxx/XXX.template. See also Table 2.1 in the Fortran documentation provided in $GALAHAD/doc/xxx.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 sha_control_type) |
specfile |
is a character string containing the name of the specification file |
void sha_analyse_matrix( struct sha_control_type* control, void **data, ipc_ *status, ipc_ n, ipc_ ne, const ipc_ row[], const ipc_ col[], ipc_ *m )
Analsyse the sparsity structure of \(H\) to generate information that will be used when estimating its values.
Parameters:
control |
is a struct whose members provide control paramters for the remaining prcedures (see sha_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:
|
n |
is a scalar variable of type ipc_, that holds the number of variables |
ne |
is a scalar variable of type ipc_, that holds the number of entries in the upper triangular part of \(H\). |
row |
is a one-dimensional array of size ne and type ipc_, that holds the row indices of the upper triangular part of \(H\). |
col |
is a one-dimensional array of size ne and type ipc_, that holds the column indices of the upper triangular part of \(H\). |
m |
is a scalar variable of type ipc_, that gives the minimum number of \((s^{(k)},y^{(k)})\) pairs that will be needed to recover a good Hessian approximation. |
void sha_recover_matrix( void **data, ipc_ *status, ipc_ ne, ipc_ m_available, ipc_ ls1, ipc_ ls2, const real_wp_ strans[][ls2], ipc_ ly1, ipc_ ly2, const real_wp_ ytrans[][ly2], real_wp_ val[], const ipc_ order[] )
Estimate the nonzero entries of the Hessian \(H\) by component-wise secant approximation.
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:
|
ne |
is a scalar variable of type ipc_, that holds the number of entries in the upper triangular part of \(H\). |
m_available |
is a scalar variable of type ipc_, that holds the number of differences provided. Ideally this will be as large as m as reported by sha_analyse_matrix, but better still there should be a further control.extra_differences to allow for unlikely singularities. |
ls1 |
is a scalar variable of type ipc_, that holds the leading (first) dimension of the array strans. |
ls2 |
is a scalar variable of type ipc_, that holds the trailing (second) dimension of the array strans. |
strans |
is a two-dimensional array of size [ls1][ls2] and type rpc_, that holds the values of the vectors \(\{s^{(k) T}\}\). Component [\(k\)][\(i\)] should hold \(s_i^{(k)}\). |
ly1 |
is a scalar variable of type ipc_, that holds the leading (first) dimension of the array ytrans. |
ly2 |
is a scalar variable of type ipc_, that holds the trailing (second) dimension of the array ytrans. |
ytrans |
is a two-dimensional array of size [ly1][ly2] and type rpc_, that holds the values of the vectors \(\{y^{(k) T}\}\). Component [\(k\)][\(i\)] should hold \(y_i^{(k)}\). |
val |
is a one-dimensional array of size ne and type rpc_, that holds the values of the entries of the upper triangular part of the symmetric matrix \(H\) in the sparse coordinate scheme. |
order |
is a one-dimensional array of size m and type ipc_, that holds the preferred order of access for the pairs \(\{(s^{(k)},y^{(k)})\}\). The \(k\)-th component of order specifies the row number of strans and ytrans that will be used as the \(k\)-th most favoured. order need not be set if the natural order, \(k, k = 1,...,\) m, is desired, and this case order should be NULL. |
void sha_information(void **data, struct sha_inform_type* inform, ipc_ *status)
Provides output information
Parameters:
data |
holds private internal data |
inform |
is a struct containing output information (see sha_inform_type) |
status |
is a scalar variable of type ipc_, that gives the exit status from the package. Possible values are (currently):
|
void sha_terminate( void **data, struct sha_control_type* control, struct sha_inform_type* inform )
Deallocate all internal private storage
Parameters:
data |
holds private internal data |
control |
is a struct containing control information (see sha_control_type) |
inform |
is a struct containing output information (see sha_inform_type) |
available structures#
sha_control_type structure#
#include <galahad_sha.h> struct sha_control_type { // fields bool f_indexing; ipc_ error; ipc_ out; ipc_ print_level; ipc_ approximation_algorithm; ipc_ dense_linear_solver; ipc_ extra_differences; ipc_ sparse_row; ipc_ recursion_max; ipc_ recursion_entries_required; bool space_critical; bool deallocate_error_fatal; char prefix[31]; };
detailed documentation#
control derived type as a C struct
components#
bool f_indexing
use C or Fortran sparse matrix indexing
ipc_ error
error and warning diagnostics occur on stream error
ipc_ out
general output occurs on stream out
ipc_ print_level
the level of output required. <= 0 gives no output, = 1 gives a one-line summary for every iteration, = 2 gives a summary of the inner iteration for each iteration, >= 3 gives increasingly verbose (debugging) output
ipc_ approximation_algorithm
which approximation algorithm should be used?
1 : unsymmetric, parallel (Alg 2.1 in paper)
2 : symmetric (Alg 2.2 in pape)
3 : composite, parallel (Alg 2.3 in paper)
4 : composite, block parallel (Alg 2.4 in paper)
ipc_ dense_linear_solver
which dense linear equation solver should be used?
1 : Gaussian elimination
2 : QR factorization
3 : singular-value decomposition
4 : singular-value decomposition with divide-and-conquer
ipc_ extra_differences
if available use an addition extra_differences differences
ipc_ sparse_row
a row is considered sparse if it has no more than .sparse_row entries
ipc_ recursion_max
limit on the maximum number of levels of recursion (Alg. 2.4)
ipc_ recursion_entries_required
the minimum number of entries in a reduced row that are required if a further level of recuresion is allowed (Alg. 2.4)
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 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’
sha_inform_type structure#
#include <galahad_sha.h> struct sha_inform_type { // fields ipc_ status; ipc_ alloc_status; ipc_ max_degree; ipc_ differences_needed; ipc_ max_reduced_degree; ipc_ approximation_algorithm_used; ipc_ bad_row; char bad_alloc[81]; };
detailed documentation#
inform derived type as a C struct
components#
ipc_ status
return status. See SHA_solve for details
ipc_ alloc_status
the status of the last attempted allocation/deallocation.
ipc_ max_degree
the maximum degree in the adgacency graph.
ipc_ differences_needed
the number of differences that will be needed.
ipc_ max_reduced_degree
the maximum reduced degree in the adgacency graph.
ipc_ approximation_algorithm_used
the approximation algorithm actually used
ipc_ bad_row
a failure occured when forming the bad_row-th row (0 = no failure).
char bad_alloc[81]
the name of the array for which an allocation/deallocation error occurred.
example calls#
This is an example of how to use the package to find a Hessian approximation using gradient differences; the code is available in $GALAHAD/src/sha/C/shat.c .
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.
/* shat.c */
/* Full test for the SHA C interface using C sparse matrix indexing */
#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <float.h>
#include "galahad_precision.h"
#include "galahad_cfunctions.h"
#include "galahad_sha.h"
int main(void) {
// Derived types
void *data;
struct sha_control_type control;
struct sha_inform_type inform;
// Set problem data
ipc_ n = 5; // dimension of H
ipc_ ne = 9; // number of entries in upper triangle of H
ipc_ m_max = 9; // upper bound on max # differences required
ipc_ row[] = {0, 0, 0, 0, 0, 1, 2, 3, 4}; // row indices, NB upper triangle
ipc_ col[] = {0, 1, 2, 3, 4, 1, 2, 3, 4}; // column indices
rpc_ val[] = {1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0}; // values
ipc_ i, j, k, l, m, status;
rpc_ rr, v;
rpc_ val_est[ne];
ipc_ ls1 = m_max, ls2 = n; // dimensions of s
ipc_ ly1 = m_max, ly2 = n; // dimensions of y
rpc_ strans[ls1][ls2];
rpc_ ytrans[ly1][ly2];
ipc_ order[m_max]; // diffference precedence order
printf(" C sparse matrix indexing\n\n");
printf(" basic tests of storage formats\n\n");
for( ipc_ algorithm=1; algorithm <= 5; algorithm++){
// Initialize SHA - use the sytr solver
sha_initialize( &data, &control, &status );
// Set user-defined control options
control.f_indexing = false; // C sparse matrix indexing
control.approximation_algorithm = algorithm;
control.extra_differences = 1;
// analyse the matrix and discover the number of differences needed
sha_analyse_matrix( &control, &data, &status, n, ne, row, col, &m );
printf(" Algorithm %" i_ipc_ " - %" i_ipc_ " differences required,"
" one extra might help\n", algorithm, m);
m = m + control.extra_differences;
srand(1);
for( ipc_ k = 0; k < m; k++) {
// set up random differences strans
for( ipc_ i = 0; i < n; i++) {
rr = ((rpc_) rand()/ RAND_MAX);
strans[k][i] = -1.0 + 2.0 * rr;
ytrans[k][i] = 0.0; // initialize ytrans as the zero matrix
}
// form y = H s
for( ipc_ l = 0; l < ne; l++) {
i = row[l];
j = col[l];
v = val[l];
ytrans[k][i] = ytrans[k][i] + v * strans[k][j];
if ( i != j ) ytrans[k][j] = ytrans[k][j] + v * strans[k][i];
}
order[k] = m - k - 1; // pick the (s,y) vectors in reverse order
}
// recover the matrix
sha_recover_matrix( &data, &status, ne, m, ls1, ls2, strans,
ly1, ly2, ytrans, val_est, order );
// ly1, ly2, ytrans, val_est, NULL );
// if the natural order is ok
printf(" H from %" i_ipc_ " differences:\n", m);
for( ipc_ i = 0; i < ne; i++) printf(" %1.2f", val_est[i]);
sha_information( &data, &inform, &status );
// Delete internal workspace
sha_terminate( &data, &control, &inform );
printf("\n\n");
}
}This is the same example, but now fortran-style indexing is used; the code is available in $GALAHAD/src/sha/C/shatf.c .
/* shatf.c */
/* Full test for the SHA C interface using Fortran sparse matrix indexing */
#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <float.h>
#include "galahad_precision.h"
#include "galahad_cfunctions.h"
#include "galahad_sha.h"
int main(void) {
// Derived types
void *data;
struct sha_control_type control;
struct sha_inform_type inform;
// Set problem data
ipc_ n = 5; // dimension of H
ipc_ ne = 9; // number of entries in upper triangle of H
ipc_ m_max = 9; // upper bound on max # differences required
ipc_ row[] = {1, 1, 1, 1, 1, 2, 3, 4, 5}; // row indices, NB upper triangle
ipc_ col[] = {1, 2, 3, 4, 5, 2, 3, 4, 5}; // column indices
rpc_ val[] = {1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0}; // values
ipc_ i, j, k, l, m, status;
rpc_ rr, v;
rpc_ val_est[ne];
ipc_ ls1 = m_max, ls2 = n; // dimensions of s
ipc_ ly1 = m_max, ly2 = n; // dimensions of y
rpc_ strans[ls1][ls2];
rpc_ ytrans[ly1][ly2];
ipc_ order[m_max]; // diffference precedence order
printf(" Fortran sparse matrix indexing\n\n");
printf(" basic tests of storage formats\n\n");
for( ipc_ algorithm=1; algorithm <= 5; algorithm++){
// Initialize SHA - use the sytr solver
sha_initialize( &data, &control, &status );
// Set user-defined control options
control.f_indexing = true; // Fortran sparse matrix indexing
control.approximation_algorithm = algorithm;
control.extra_differences = 1;
// analyse the matrix and discover the number of differences needed
sha_analyse_matrix( &control, &data, &status, n, ne, row, col, &m );
printf(" Algorithm %" i_ipc_ " - %" i_ipc_ " differences required,"
" one extra might help\n", algorithm, m);
m = m + control.extra_differences;
srand(1);
for( ipc_ k = 0; k < m; k++) {
// set up random differences strans
for( ipc_ i = 0; i < n; i++) {
rr = ((rpc_) rand()/ RAND_MAX);
strans[k][i] = -1.0 + 2.0 * rr;
ytrans[k][i] = 0.0; // initialize ytrans as the zero matrix
}
// form y = H s
for( ipc_ l = 0; l < ne; l++) {
i = row[l]-1;
j = col[l]-1;
v = val[l];
ytrans[k][i] = ytrans[k][i] + v * strans[k][j];
if ( i != j ) ytrans[k][j] = ytrans[k][j] + v * strans[k][i];
}
order[k] = m - k; // pick the (s,y) vectors in reverse order
}
// recover the matrix
sha_recover_matrix( &data, &status, ne, m, ls1, ls2, strans,
ly1, ly2, ytrans, val_est, order );
// ly1, ly2, ytrans, val_est, NULL );
// if the natural order is ok
printf(" H from %" i_ipc_ " differences:\n", m);
for( ipc_ i = 0; i < ne; i++) printf(" %1.2f", val_est[i]);
sha_information( &data, &inform, &status );
// Delete internal workspace
sha_terminate( &data, &control, &inform );
printf("\n\n");
}
}