NAG Library Function Document
nag_zgbsvx (f07bpc)
1
Purpose
nag_zgbsvx (f07bpc) uses the
factorization to compute the solution to a complex system of linear equations
where
is an
by
band matrix with
subdiagonals and
superdiagonals, and
and
are
by
matrices. Error bounds on the solution and a condition estimate are also provided.
2
Specification
#include <nag.h> |
#include <nagf07.h> |
void |
nag_zgbsvx (Nag_OrderType order,
Nag_FactoredFormType fact,
Nag_TransType trans,
Integer n,
Integer kl,
Integer ku,
Integer nrhs,
Complex ab[],
Integer pdab,
Complex afb[],
Integer pdafb,
Integer ipiv[],
Nag_EquilibrationType *equed,
double r[],
double c[],
Complex b[],
Integer pdb,
Complex x[],
Integer pdx,
double *rcond,
double ferr[],
double berr[],
double *recip_growth_factor,
NagError *fail) |
|
3
Description
nag_zgbsvx (f07bpc) performs the following steps:
1. |
Equilibration
The linear system to be solved may be badly scaled. However, the system can be equilibrated as a first stage by setting . In this case, real scaling factors are computed and these factors then determine whether the system is to be equilibrated. Equilibrated forms of the systems , and are
and
respectively, where and are diagonal matrices, with positive diagonal elements, formed from the computed scaling factors.
When equilibration is used, will be overwritten by and will be overwritten by (or when the solution of or is sought). |
2. |
Factorization
The matrix , or its scaled form, is copied and factored using the decomposition
where is a permutation matrix, is a unit lower triangular matrix, and is upper triangular.
This stage can be by-passed when a factored matrix (with scaled matrices and scaling factors) are supplied; for example, as provided by a previous call to nag_zgbsvx (f07bpc) with the same matrix . |
3. |
Condition Number Estimation
The factorization of determines whether a solution to the linear system exists. If some diagonal element of is zero, then is exactly singular, no solution exists and the function returns with a failure. Otherwise the factorized form of is used to estimate the condition number of the matrix . If the reciprocal of the condition number is less than machine precision then a warning code is returned on final exit. |
4. |
Solution
The (equilibrated) system is solved for ( or ) using the factored form of (). |
5. |
Iterative Refinement
Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for the computed solution. |
6. |
Construct Solution Matrix
If equilibration was used, the matrix is premultiplied by (if ) or (if or ) so that it solves the original system before equilibration. |
4
References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5
Arguments
- 1:
– Nag_OrderTypeInput
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_FactoredFormTypeInput
-
On entry: specifies whether or not the factorized form of the matrix
is supplied on entry, and if not, whether the matrix
should be equilibrated before it is factorized.
- afb and ipiv contain the factorized form of . If , the matrix has been equilibrated with scaling factors given by r and c. ab, afb and ipiv are not modified.
- The matrix will be copied to afb and factorized.
- The matrix will be equilibrated if necessary, then copied to afb and factorized.
Constraint:
, or .
- 3:
– Nag_TransTypeInput
-
On entry: specifies the form of the system of equations.
- (No transpose).
- (Transpose).
- (Conjugate transpose).
Constraint:
, or .
- 4:
– IntegerInput
-
On entry: , the number of linear equations, i.e., the order of the matrix .
Constraint:
.
- 5:
– IntegerInput
-
On entry: , the number of subdiagonals within the band of the matrix .
Constraint:
.
- 6:
– IntegerInput
-
On entry: , the number of superdiagonals within the band of the matrix .
Constraint:
.
- 7:
– IntegerInput
-
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
Constraint:
.
- 8:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
ab
must be at least
.
On entry: the
by
coefficient matrix
.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements
, for row
and column
, depends on the
order argument as follows:
- if , is stored as ;
- if , is stored as .
See
Section 9 for further details.
If
and
,
must have been equilibrated by the scaling factors in
r and/or
c.
On exit: if
or
, or if
and
,
ab is not modified.
If
then, if no constraints are violated,
is scaled as follows:
- if , ;
- if , ;
- if , .
- 9:
– IntegerInput
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix
in the array
ab.
Constraint:
.
- 10:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
afb
must be at least
.
On entry: if
or
,
afb need not be set.
If
, details of the
factorization of the
by
band matrix
, as computed by
nag_zgbtrf (f07brc).
The elements, , of the upper triangular band factor with super-diagonals, and the multipliers, , used to form the lower triangular factor are stored. The elements , for and , and , for and , are stored where is stored on entry.
If
,
afb is the factorized form of the equilibrated matrix
.
On exit: if
,
afb is unchanged from entry.
Otherwise, if no constraints are violated, then if
,
afb returns details of the
factorization of the band matrix
, and if
,
afb returns details of the
factorization of the equilibrated band matrix
(see the description of
ab for the form of the equilibrated matrix).
- 11:
– IntegerInput
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix
in the array
afb.
Constraint:
.
- 12:
– IntegerInput/Output
-
Note: the dimension,
dim, of the array
ipiv
must be at least
.
On entry: if
or
,
ipiv need not be set.
If
,
ipiv contains the pivot indices from the factorization
, as computed by
nag_dgbtrf (f07bdc); row
of the matrix was interchanged with row
.
On exit: if
,
ipiv is unchanged from entry.
Otherwise, if no constraints are violated,
ipiv contains the pivot indices that define the permutation matrix
; at the
th step row
of the matrix was interchanged with row
.
indicates a row interchange was not required.
If , the pivot indices are those corresponding to the factorization of the original matrix .
If , the pivot indices are those corresponding to the factorization of of the equilibrated matrix .
- 13:
– Nag_EquilibrationType *Input/Output
-
On entry: if
or
,
equed need not be set.
If
,
equed must specify the form of the equilibration that was performed as follows:
- if , no equilibration;
- if , row equilibration, i.e., has been premultiplied by ;
- if , column equilibration, i.e., has been postmultiplied by ;
- if , both row and column equilibration, i.e., has been replaced by .
On exit: if
,
equed is unchanged from entry.
Otherwise, if no constraints are violated,
equed specifies the form of equilibration that was performed as specified above.
Constraint:
if , , , or .
- 14:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
r
must be at least
.
On entry: if
or
,
r need not be set.
If
and
or
,
r must contain the row scale factors for
,
; each element of
r must be positive.
On exit: if
,
r is unchanged from entry.
Otherwise, if no constraints are violated and
or
,
r contains the row scale factors for
,
, such that
is multiplied on the left by
; each element of
r is positive.
- 15:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
c
must be at least
.
On entry: if
or
,
c need not be set.
If
and
or
,
c must contain the column scale factors for
,
; each element of
c must be positive.
On exit: if
,
c is unchanged from entry.
Otherwise, if no constraints are violated and
or
,
c contains the row scale factors for
,
; each element of
c is positive.
- 16:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
b
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by right-hand side matrix .
On exit: if
,
b is not modified.
If
and
or
,
b is overwritten by
.
If
or
and
or
,
b is overwritten by
.
- 17:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 18:
– ComplexOutput
-
Note: the dimension,
dim, of the array
x
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On exit: if
NE_NOERROR or
NE_SINGULAR_WP, the
by
solution matrix
to the original system of equations. Note that the arrays
and
are modified on exit if
, and the solution to the equilibrated system is
if
and
or
, or
if
or
and
or
.
- 19:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraints:
- if ,
;
- if , .
- 20:
– double *Output
-
On exit: if no constraints are violated, an estimate of the reciprocal condition number of the matrix (after equilibration if that is performed), computed as .
- 21:
– doubleOutput
-
On exit: if
NE_NOERROR or
NE_SINGULAR_WP, an estimate of the forward error bound for each computed solution vector, such that
where
is the
th column of the computed solution returned in the array
x and
is the corresponding column of the exact solution
. The estimate is as reliable as the estimate for
rcond, and is almost always a slight overestimate of the true error.
- 22:
– doubleOutput
-
On exit: if
NE_NOERROR or
NE_SINGULAR_WP, an estimate of the component-wise relative backward error of each computed solution vector
(i.e., the smallest relative change in any element of
or
that makes
an exact solution).
- 23:
– double *Output
-
On exit: if
NE_NOERROR, the reciprocal pivot growth factor
, where
denotes the maximum absolute element norm. If
, the stability of the
factorization of (equilibrated)
could be poor. This also means that the solution
x, condition estimate
rcond, and forward error bound
ferr could be unreliable. If the factorization fails with
NE_SINGULAR, then
contains the reciprocal pivot growth factor for the leading
columns of
.
- 24:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INT_3
-
On entry, , and .
Constraint: .
On entry, , and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_SINGULAR
-
Element of the diagonal is exactly zero.
The factorization has been completed, but the factor is exactly singular, so the solution and error bounds could not be computed.
is returned.
- NE_SINGULAR_WP
-
is nonsingular, but
rcond is less than
machine precision, meaning that the matrix is singular to working precision.
Nevertheless, the solution and error bounds are computed because there
are a number of situations where the computed solution can be more accurate
than the value of
rcond would suggest.
7
Accuracy
For each right-hand side vector
, the computed solution
is the exact solution of a perturbed system of equations
, where
is a modest linear function of
, and
is the
machine precision. See Section 9.3 of
Higham (2002) for further details.
If
is the true solution, then the computed solution
satisfies a forward error bound of the form
where
.
If
is the
th column of
, then
is returned in
and a bound on
is returned in
. See Section 4.4 of
Anderson et al. (1999) for further details.
8
Parallelism and Performance
nag_zgbsvx (f07bpc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zgbsvx (f07bpc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The band storage scheme for the array
ab is illustrated by the following example, when
,
, and
. Storage of the band matrix
in the array
ab:
The total number of floating-point operations required to solve the equations
depends upon the pivoting required, but if
then it is approximately bounded by
for the factorization and
for the solution following the factorization. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization. The solution is then refined, and the errors estimated, using iterative refinement; see
nag_zgbrfs (f07bvc) for information on the floating-point operations required.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of
Higham (2002) for further details.
The real analogue of this function is
nag_dgbsvx (f07bbc).
10
Example
This example solves the equations
where
is the band matrix
and
Estimates for the backward errors, forward errors, condition number and pivot growth are also output, together with information on the equilibration of .
10.1
Program Text
Program Text (f07bpce.c)
10.2
Program Data
Program Data (f07bpce.d)
10.3
Program Results
Program Results (f07bpce.r)