GALAHAD NREK package#
purpose#
The nrek package uses an extended-Krylov-subspace iteration to find the
global minimizer of a norm-regularized quadratic objective function;
this is commonly known as the norm-regularization subproblem.
The aim is to minimize the regularized quadratic objective function
Factorization of the matrices \(H\) and, if present, \(S\)
will be required, so this package is most suited
for the case where such a factorization may be found efficiently. If
this is not the case, the package glrt may be preferred.
See Section 4 of $GALAHAD/doc/nrek.pdf for additional details.
method#
The required solution \(x_*\) necessarily satisfies the optimality condition \(H x_* + \lambda_* S x_* + c = 0\), where \(\lambda_* \sigma \|x_*\|^{p-2}\) is a Lagrange multiplier for the regularization. In addition in all cases, the matrix \(H + \lambda_* S\) will be positive semi-definite; in most instances it will actually be positive definite, but in special “hard” cases singularity is a possibility.
The method is iterative, and is based upon building a solution approximation from an orthogonal basis of the evolving extended Krylov subspaces \({\cal K}_{2m+1}(H,c) = \mbox{span}\{c,H^{-1}c,H c,H^{-2}c,H^2c,\ldots,\) \(H^{-m}c,H^{m}c\}\) as \(m\) increases. The key observations are (i) the manifold of solutions to the optimality system \[ ( H + \lambda I ) x(\lambda) = - c\] as a function of \(\sigma\) is of approximately very low rank, (ii) the subspace \({\cal K}_{2m+1}(H,c)\) rapidly gives a very good approximation to this manifold, (iii) it is straightforward to build an orthogonal basis of \({\cal K}_{2m+1}(H,c)\) using short-term recurrences and a single factorization of \(H\), and (iv) solutions to the norm-regularization subproblem restricted to elements of the orthogonal subspace may be found very efficiently using effective high-order root-finding methods. Coping with general scalings \(S\) is a straightforward extension so long as factorization of \(S\) is also possible.
reference#
The method is described in detail in
H. Al Daas and N. I. M. Gould. Extended-Krylov-subspace methods for trust-region and norm-regularization subproblems. Preprint STFC-P-2025-002, Rutherford Appleton Laboratory, Oxfordshire, England.
matrix storage#
symmetric storage#
The symmetric \(n\) by \(n\) matrices \(H\) and, optionally, \(S\) 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).
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 \(1 \leq j \leq i \leq n\)) need be held. In this case the lower triangle should be stored by rows, that is component \((i-1) * i / 2 + j\) of the storage array H_val will hold the value \(H_{ij}\) (and, by symmetry, \(H_{ji}\)) for \(1 \leq j \leq i \leq n\). 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, \(1 \leq l \leq ne\), of \(H\), its row index i, column index j and value \(H_{ij}\), \(1 \leq j \leq i \leq n\), 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+1) holds the total number of entries plus one. The column indices j, \(1 \leq j \leq i\), 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 \(1 \leq i \neq j \leq n\)) only the diagonals entries \(H_{ii}\), \(1 \leq i \leq n\) 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 nrek package must be called in the following order:
nrek_initialize - provide default control parameters and set up initial data structures
nrek_read_specfile (optional) - override control values by reading replacement values from a file
nrek_import - set up problem data structures and fixed values
nrek_import_s - (optional) set up problem data structures and fixed values for the scaling matrix \(S\), if any
nrek_reset_control (optional) - possibly change control parameters if a sequence of problems are being solved
nrek_solve_problem - solve the trust-region problem
nrek_information (optional) - recover information about the solution and solution process
nrek_terminate - deallocate data structures
See the examples section for illustrations of use.
parametric real type T and integer type INT#
Below, the symbol T refers to a parametric real type that may be Float32 (single precision), Float64 (double precision) or, if supported, Float128 (quadruple precision). The symbol INT refers to a parametric integer type that may be Int32 (32-bit integer) or Int64 (64-bit integer).
callable functions#
function nrek_initialize(T, INT, data, control, status)
Set default control values and initialize private data
Parameters:
data |
holds private internal data |
control |
is a structure containing control information (see nrek_control_type) |
status |
is a scalar variable of type INT that gives the exit status from the package. Possible values are (currently):
|
function nrek_read_specfile(T, INT, control, 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/nrek/NREK.template. See also Table 2.1 in the Fortran documentation provided in $GALAHAD/doc/nrek.pdf for a list of how these keywords relate to the components of the control structure.
Parameters:
control |
is a structure containing control information (see nrek_control_type) |
specfile |
is a one-dimensional array of type Vararg{Cchar} that must give the name of the specification file |
function nrek_import(T, INT, control, data, status, n, H_type, H_ne, H_row, H_col, H_ptr)
Import problem data into internal storage prior to solution.
Parameters:
control |
is a structure whose members provide control parameters for the remaining procedures (see nrek_control_type) |
data |
holds private internal data |
status |
is a scalar variable of type INT that gives the exit status from the package. Possible values are:
|
n |
is a scalar variable of type INT that holds the number of rows (and columns) of H. |
H_type |
is a one-dimensional array of type Vararg{Cchar} that specifies the symmetric storage scheme used for the Hessian, \(H\). It should be one of ‘coordinate’, ‘sparse_by_rows’, ‘dense’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’; lower or upper case variants are allowed. |
H_ne |
is a scalar variable of type INT 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 INT 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 C_NULL. |
H_col |
is a one-dimensional array of size H_ne and type INT 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 or diagonal storage schemes are used, and in this case can be C_NULL. |
H_ptr |
is a one-dimensional array of size n+1 and type INT 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 C_NULL. |
function nrek_import_s(T, INT, data, status, n, S_type, S_ne, S_row, S_col, S_ptr)
Import data for the scaling matrix \(S\) into internal storage prior to solution.
Parameters:
data |
holds private internal data |
status |
is a scalar variable of type INT that gives the exit status from the package. Possible values are:
|
n |
is a scalar variable of type INT that holds the number of rows (and columns) of \(S\). |
S_type |
is a one-dimensional array of type Vararg{Cchar} that specifies the symmetric storage scheme used for the scaling matrix, \(S\). It should be one of ‘coordinate’, ‘sparse_by_rows’, ‘dense’, ‘diagonal’, ‘scaled_identity’ or ‘identity’; lower or upper case variants are allowed. |
S_ne |
is a scalar variable of type INT that holds the number of entries in the lower triangular part of \(S\) in the sparse co-ordinate storage scheme. It need not be set for any of the other schemes. |
S_row |
is a one-dimensional array of size S_ne and type INT that holds the row indices of the lower triangular part of \(S\) 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 C_NULL. |
S_col |
is a one-dimensional array of size S_ne and type INT that holds the column indices of the lower triangular part of \(S\) in either the sparse co-ordinate, or the sparse row-wise storage scheme. It need not be set when the dense, diagonal or identity storage schemes are used, and in this case can be C_NULL. |
S_ptr |
is a one-dimensional array of size n+1 and type INT that holds the starting position of each row of the lower triangular part of \(S\), 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 C_NULL. |
function nrek_reset_control(T, INT, control, data, status)
Reset control parameters after import if required.
Parameters:
control |
is a structure whose members provide control parameters for the remaining procedures (see nrek_control_type) |
data |
holds private internal data |
status |
is a scalar variable of type INT that gives the exit status from the package. Possible values are:
|
function nrek_solve_problem(T, INT, data, status, n, H_ne, H_val, c, power, weight, x, S_ne, S_val)
Solve the norm-regularization subproblem.
Parameters:
data |
holds private internal data |
status |
is a scalar variable of type INT that gives the entry and exit status from the package. On initial entry, status must be set to 1. Possible exit values are:
|
n |
is a scalar variable of type INT that holds the number of variables |
H_ne |
is a scalar variable of type INT that holds the number of entries in the lower triangular part of the Hessian matrix \(H\). |
H_val |
is a one-dimensional array of size h_ne and type T that holds the values of the entries of the lower triangular part of the Hessian matrix \(H\) in any of the available storage schemes. |
c |
is a one-dimensional array of size n and type T that holds the linear term \(c\) of the objective function. The j-th component of |
power |
is a scalar of type T that holds the regularization power, \(p\), used. power must be strictly larger than two |
weight |
is a scalar of type T that holds the regularization weight, \(\Delta\), used. weight must be strictly positive |
x |
is a one-dimensional array of size n and type T that holds the values \(x\) of the optimization variables. The j-th component of |
S_ne |
is a scalar variable of type INT that holds the number of entries in the scaling matrix \(S\) if it not the identity matrix. |
S_val |
is a one-dimensional array of size S_ne and type T that holds the values of the entries of the scaling matrix \(S\), if it is not the identity matrix, in any of the available storage schemes. If S_val is C_NULL, \(S\) will be taken to be the identity matrix. |
function nrek_information(T, INT, data, inform, status)
Provides output information
Parameters:
data |
holds private internal data |
inform |
is a structure containing output information (see nrek_inform_type) |
status |
is a scalar variable of type INT that gives the exit status from the package. Possible values are (currently):
|
function nrek_terminate(T, INT, data, control, inform)
Deallocate all internal private storage
Parameters:
data |
holds private internal data |
control |
is a structure containing control information (see nrek_control_type) |
inform |
is a structure containing output information (see nrek_inform_type) |
available structures#
nrek_control_type structure#
struct nrek_control_type{T,INT} f_indexing::Bool error::INT out::INT print_level::INT eks_max::INT it_max::INT f::T increase::T stop_residual::T reorthogonalize::Bool s_version_52::Bool perturb_c::Bool stop_check_all_orders::Bool new_weight::Bool new_values::Bool space_critical::Bool deallocate_error_fatal::Bool linear_solver::NTuple{31,Cchar} linear_solver_for_s::NTuple{31,Cchar} prefix::NTuple{31,Cchar} sls_control::sls_control_type{T,INT} sls_s_control::sls_control_type{T,INT} rqs_control::ir_control_type{T,INT}
detailed documentation#
control derived type as a Julia structure
components#
Bool f_indexing
use C or Fortran sparse matrix indexing
INT error
unit for error messages
INT out
unit for monitor output
INT print_level
controls level of diagnostic output
INT eks_max
maximum dimension of the extended Krylov space employed. If a negative value is given, the value 100 will be used instead
INT it_max
the maximum number of iterations allowed. If a negative value is given, the value 100 will be used instead
T f
the value of \(f\) in the objective function. This value has no effect on the computed \(x\), and takes the value 0.0 by default
T increase
the value of the increase factor for a suggested subsequent regularization weight, see control[‘next_weight’]. The suggested weight will be increase times the current weight
T stop_residual
the value of the stopping tolerance used by the algorithm. The iteration stops as soon as \(x\) and \(\lambda\) are found to satisfy \(\| ( H + \lambda S ) x + c \| <\) stop_residual \(\times \max( 1, \|c\| )\)
Bool reorthogonalize
should be set to true if the generated basis of the extended-Krylov subspace is to be reorthogonalized at every iteration. This can be very expensive, and is generally not warranted
Bool s_version_52
should be set to true if Algorithm 5.2 in the paper is used to generate the extended Krylov space recurrences when a non-unit \(S\) is given, and false if those from Algorithm B.3 ares used instead. In practice, there is very little difference in performance and accuracy
Bool perturb_c
should be set to true if the user wishes to make a tiny pseudo-random perturbations to the components of the term \(c\) to try to protect from the so-called (probability zero) “hard” case. Perturbations are generally not needed, and should only be used in very exceptional cases
Bool stop_check_all_orders
should be set to true if the algorithm checks for termination for each new member of the extended Krylov space. Such checks incur some extra cost, and experience shows that testing every second member is sufficient
Bool new_weight
should be set to true if the call retains the previous \(H\), \(S\) and \(c\), but with a new, smaller weight
Bool new_values
should be set to true if the any of the values of \(H\), \(S\) and \(c\) has changed since a previous call
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 linear_solver[31]
the name of the linear equation solver used to solve any symmetric positive-definite linear system involving \(H\) that might arise. Possible choices are currently: ‘sils’, ‘ma27’, ‘ma57’, ‘ma77’, ‘ma86’, ‘ma87’, ‘ma97’, ssids, ‘pardiso’, ‘wsmp’, ‘sytr’, ‘potr’ and ‘pbtr’ although only ‘sytr’, ‘potr’, ‘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 linear_solver_for_s[31]
the name of the linear equation solver used to solve any symmetric positive-definite linear system involving the optional \(S\) that might arise. Possible choices are currently: ‘sils’, ‘ma27’, ‘ma57’, ‘ma77’, ‘ma86’, ‘ma87’, ‘ma97’, ssids, ‘pardiso’, ‘wsmp’, ‘sytr’, ‘potr’ and ‘pbtr’ although only ‘sytr’, ‘potr’, ‘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.
NTuple{31,Cchar} prefix
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 the Cholesky factorization and solution involving \(H\) (see sls_c documentation)
struct sls_control_type sls_s_control
control parameters for the Cholesky factorization and solution involving \(S\) (see sls_c documentation)
struct rqs_control_type rqs_control
control parameters for the solution of diagonal norm-regularization subproblems (see rqs_c documentation)
nrek_time_type structure#
struct nrek_time_type{T} total::T assemble::T analyse::T factorize::T solve::T clock_total::T clock_assemble::T clock_analyse::T clock_factorize::T clock_solve::T
detailed documentation#
time derived type as a Julia structure
components#
T total
total CPU time spent in the package
T assemble
CPU time spent building \(H\) and \(S\).
T analyse
CPU time spent reordering \(H\) and \(S\) prior to factorization.
T factorize
CPU time spent factorizing \(H\) and \(S\).
T solve
CPU time spent solving linear systems inolving \(H\) and \(S\).
T clock_total
total clock time spent in the package
T clock_assemble
clock time spent building \(H\) and \(S\)
T clock_analyse
clock time spent reordering \(H\) and \(S\) prior to factorization
T clock_factorize
clock time spent factorizing \(H\) and \(S\)
T clock_solve
clock time spent solving linear systems inolving \(H\) and \(S\)
nrek_inform_type structure#
struct nrek_inform_type{T,INT} status::INT alloc_status::INT iter::INT n_vec::INT obj::T x_norm::T multiplier::T weight::T next_weight::T error::T bad_alloc::NTuple{81,Cchar} time::nrek_time_type{T} sls_inform::sls_inform_type{T,INT} sls_s_inform::sls_inform_type{T,INT} rqs_inform::rqs_inform_type{T,INT}
detailed documentation#
inform derived type as a Julia structure
components#
INT status
reported return status:
0
the solution has been found
-1
an array allocation has failed
-2
an array deallocation has failed
-3
n and/or \(\Delta\) is not positive, or the requirement that type contains its relevant string ‘dense’, ‘coordinate’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated.
-9
the analysis phase of the factorization failed; the return status from the factorization package is given by inform.sls_inform.status or inform.sls_s_inform.status as appropriate
-10
the factorization failed; the return status from the factorization package is given by inform.sls_inform.status or inform.sls_s_inform.status as appropriate
-11
the solution of a set of linear equations using factors from the factorization package failed; the return status from the factorization package is given by inform.sls_inform.status or inform.sls_s_inform.status as appropriate
-15
\(S\) does not appear to be strictly diagonally dominant
-16
ill-conditioning has prevented further progress
-18
too many iterations have been required. This may happen if control.eks max is too small, but may also be symptomatic of a badly scaled problem.
-31
a resolve call has been made before an initial call (see control.new_weight and control.new_values)
-38
an error occurred in a call to an LAPACK subroutine
INT alloc_status
STAT value after allocate failure.
INT iter
the total number of iterations required
INT n_vec
the number of orthogonal vectors required (the dimension of the extended-Krylov subspace)
T obj
the value of the quadratic function \(f + c^T x + \frac{1}{2} x^T H x\)
T obj_regularized
the value of the regularized quadratic function \(r(x)\)
T x_norm
the \(S\) -norm of \(x\), \(||x||_S\)
T multiplier
the Lagrange multiplier associated with the regularization
T weight
the value of the current weight
T next_weight
the value of the the proposed next weight to be used if the current weight proves to be too small (see inform.increase).
T error
the value of the norm of the maximum relative residual error, \(\|(H+\lambda S) x + c\|/\max(1,\|c\|)\)
NTuple{81,Cchar} bad_alloc
name of array that provoked an allocate failure
struct nrek_time_type time
time information
struct sls_inform_type sls_inform
Cholesky information for factorization and solves with \(H\) (see sls_c documentation)
struct sls_inform_type sls_s_inform
Cholesky information for factorization and solves with \(S\) (see sls_c documentation)
struct rqs_inform_type rqs_inform
diagonal norm-regularization solve information (see rqs_c documentation)
example calls#
This is an example of how to use the package to solve a trust-region subproblem; the code is available in $GALAHAD/src/nrek/Julia/test_nrek.jl . A variety of supported Hessian and scaling matrix storage formats are shown.
# test_nrek.jl
# Simple code to test the Julia interface to NREK
using GALAHAD
using Test
using Printf
using Accessors
using Quadmath
function test_nrek(::Type{T}, ::Type{INT}; dls::String="pbtr") where {T,INT}
# Derived types
data = Ref{Ptr{Cvoid}}()
control = Ref{nrek_control_type{T,INT}}()
inform = Ref{nrek_inform_type{T,INT}}()
# Set problem data
n = INT(3) # dimension of H
m = INT(1) # dimension of A
H_ne = INT(4) # number of elements of H
S_ne = INT(3) # number of elements of M
H_dense_ne = INT(6) # number of elements of H
S_dense_ne = INT(6) # number of elements of M
H_row = INT[1, 2, 3, 3] # row indices, NB lower triangle
H_col = INT[1, 2, 3, 1]
H_ptr = INT[1, 2, 3, 5]
S_row = INT[1, 2, 3] # row indices, NB lower triangle
S_col = INT[1, 2, 3]
S_ptr = INT[1, 2, 3, 4]
H_val = T[1.0, 2.0, 3.0, 4.0]
S_val = T[1.0, 2.0, 1.0]
H_dense = T[1.0, 0.0, 2.0, 4.0, 0.0, 3.0]
S_dense = T[1.0, 0.0, 2.0, 0.0, 0.0, 1.0]
H_diag = T[1.0, 0.0, 2.0]
S_diag = T[1.0, 2.0, 1.0]
c = T[0.0, 2.0, 0.0]
power = T(3.0) # regularization power
st = ' '
status = Ref{INT}()
x = zeros(T, n)
sr = ""
@printf(" Fortran sparse matrix indexing\n\n")
@printf(" basic tests of storage formats\n\n")
for s_is in 0:1 # include a scaling matrix?
for storage_type in 1:4
# Initialize NREK
nrek_initialize(T, INT, data, control, status)
# Linear solvers
@reset control[].linear_solver = galahad_linear_solver(dls)
@reset control[].linear_solver_for_s = galahad_linear_solver(dls)
# sparse co-ordinate storage
if storage_type == 1
st = 'C'
# import the control parameters and structural data
nrek_import(T, INT, control, data, status, n,
"coordinate", H_ne, H_row, H_col, C_NULL)
if s_is == 1
nrek_s_import(T, INT, data, status, n,
"coordinate", S_ne, S_row, S_col, C_NULL)
end
end
# sparse by rows
if storage_type == 2
st = 'R'
# import the control parameters and structural data
nrek_import(T, INT, control, data, status, n,
"sparse_by_rows", H_ne, C_NULL, H_col, H_ptr)
if s_is == 1
nrek_s_import(T, INT, data, status, n,
"sparse_by_rows", S_ne, C_NULL, S_col, S_ptr)
end
end
# dense
if storage_type == 3
st = 'D'
# import the control parameters and structural data
nrek_import(T, INT, control, data, status, n,
"dense", H_ne, C_NULL, C_NULL, C_NULL)
if s_is == 1
nrek_s_import(T, INT, data, status, n,
"dense", S_ne, C_NULL, C_NULL, C_NULL)
end
end
# diagonal
if storage_type == 4
st = 'L'
# import the control parameters and structural data
nrek_import(T, INT, control, data, status, n,
"diagonal", H_ne, C_NULL, C_NULL, C_NULL)
if s_is == 1
nrek_s_import(T, INT, data, status, n,
"diagonal", S_ne, C_NULL, C_NULL, C_NULL)
end
end
for r_is in 1:2 # original or larger weight
if (r_is == 1)
weight = one(T)
else
weight = inform[].next_weight
@reset control[].new_weight = true
nrek_reset_control(T, INT, control, data, status)
end
if (r_is == 2) && (s_is == 1)
sr = "S-"
elseif r_is == 2
sr = "- "
elseif s_is == 1
sr = "S "
else
sr = " "
end
# solve the problem
# sparse co-ordinate storage
if storage_type == 1
if s_is == 1
nrek_solve_problem(T, INT, data, status, n,
H_ne, H_val, c, power, weight, x,
S_ne, S_val, 0, 0, C_NULL, C_NULL)
else
nrek_solve_problem(T, INT, data, status, n,
H_ne, H_val, c, power, weight, x,
0, C_NULL, 0, 0, C_NULL, C_NULL)
end
end
# sparse by rows
if storage_type == 2
if s_is == 1
nrek_solve_problem(T, INT, data, status, n,
H_ne, H_val, c, power, weight, x,
S_ne, S_val, 0, 0, C_NULL, C_NULL)
else
nrek_solve_problem(T, INT, data, status, n,
H_ne, H_val, c, power, weight, x,
0, C_NULL, 0, 0, C_NULL, C_NULL)
end
end
# dense
if storage_type == 3
if s_is == 1
nrek_solve_problem(T, INT, data, status, n,
H_dense_ne, H_dense_val, c, power, weight, x,
S_dense_ne, S_dense, 0, 0, C_NULL, C_NULL)
else
nrek_solve_problem(T, INT, data, status, n,
H_dense_ne, H_dense_val, c, power, weight, x,
0, C_NULL, 0, 0, C_NULL, C_NULL)
end
end
# diagonal
if storage_type == 4
if s_is == 1
nrek_solve_problem(T, INT, data, status, n,
n, H_diag_val, c, power, weight, x,
n, S_diag, 0, 0, C_NULL, C_NULL)
else
nrek_solve_problem(T, INT, data, status, n,
n, H_diag_val, c, power, weight, x,
0, C_NULL, 0, 0, C_NULL, C_NULL)
end
end
nrek_information(T, INT, data, inform, status)
@printf("format %c%s: NREK_solve_problem exit status = %1i,
f = %.2f\n", st, sr, inform[].status, inform[].obj)
end
# Delete internal workspace
nrek_terminate(T, INT, data, control, inform)
end
end
return 0
end
for (T, INT, libgalahad) in ((Float32 , Int32, GALAHAD.libgalahad_single ),
(Float32 , Int64, GALAHAD.libgalahad_single_64 ),
(Float64 , Int32, GALAHAD.libgalahad_double ),
(Float64 , Int64, GALAHAD.libgalahad_double_64 ),
(Float128, Int32, GALAHAD.libgalahad_quadruple ),
(Float128, Int64, GALAHAD.libgalahad_quadruple_64))
if isfile(libgalahad)
@testset "NREK -- $T -- $INT" begin
@test test_nrek(T, INT) == 0
end
end