NAG CL Interface
f07bsc (zgbtrs)

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1 Purpose

f07bsc solves a complex band system of linear equations with multiple right-hand sides,
AX=B ,  ATX=B   or   AHX=B ,  
where A has been factorized by f07brc.

2 Specification

#include <nag.h>
void  f07bsc (Nag_OrderType order, Nag_TransType trans, Integer n, Integer kl, Integer ku, Integer nrhs, const Complex ab[], Integer pdab, const Integer ipiv[], Complex b[], Integer pdb, NagError *fail)
The function may be called by the names: f07bsc, nag_lapacklin_zgbtrs or nag_zgbtrs.

3 Description

f07bsc is used to solve a complex band system of linear equations AX=B, ATX=B or AHX=B, the function must be preceded by a call to f07brc which computes the LU factorization of A as A=PLU. The solution is computed by forward and backward substitution.
If trans=Nag_NoTrans, the solution is computed by solving PLY=B and then UX=Y.
If trans=Nag_Trans, the solution is computed by solving UTY=B and then LTPTX=Y.
If trans=Nag_ConjTrans, the solution is computed by solving UHY=B and then LHPTX=Y.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: order Nag_OrderType Input
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 order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: trans Nag_TransType Input
On entry: indicates the form of the equations.
trans=Nag_NoTrans
AX=B is solved for X.
trans=Nag_Trans
ATX=B is solved for X.
trans=Nag_ConjTrans
AHX=B is solved for X.
Constraint: trans=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
3: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
4: kl Integer Input
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
5: ku Integer Input
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
6: nrhs Integer Input
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
7: ab[dim] const Complex Input
Note: the dimension, dim, of the array ab must be at least max(1,pdab×n).
On entry: the LU factorization of A, as returned by f07brc.
8: pdab Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array ab.
Constraint: pdab2×kl+ku+1.
9: ipiv[dim] const Integer Input
Note: the dimension, dim, of the array ipiv must be at least max(1,n).
On entry: the pivot indices, as returned by f07brc.
10: b[dim] Complex Input/Output
Note: the dimension, dim, of the array b must be at least
  • max(1,pdb×nrhs) when order=Nag_ColMajor;
  • max(1,n×pdb) when order=Nag_RowMajor.
The (i,j)th element of the matrix B is stored in
  • b[(j-1)×pdb+i-1] when order=Nag_ColMajor;
  • b[(i-1)×pdb+j-1] when order=Nag_RowMajor.
On entry: the n×r right-hand side matrix B.
On exit: the n×r solution matrix X.
11: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax(1,n);
  • if order=Nag_RowMajor, pdbmax(1,nrhs).
12: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, kl=value.
Constraint: kl0.
On entry, ku=value.
Constraint: ku0.
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pdab=value.
Constraint: pdab>0.
On entry, pdb=value.
Constraint: pdb>0.
NE_INT_2
On entry, pdb=value and n=value.
Constraint: pdbmax(1,n).
On entry, pdb=value and nrhs=value.
Constraint: pdbmax(1,nrhs).
NE_INT_3
On entry, pdab=value, kl=value and ku=value.
Constraint: pdab2×kl+ku+1.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of equations (A+E)x=b, where
|E|c(k)ε|L||U| ,  
c(k) is a modest linear function of k=kl+ku+1, and ε is the machine precision. This assumes kn.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x c(k)cond(A,x)ε  
where cond(A,x)=|A-1||A||x|/xcond(A)=|A-1||A|κ(A).
Note that cond(A,x) can be much smaller than cond(A), and cond(AH) (which is the same as cond(AT)) can be much larger (or smaller) than cond(A).
Forward and backward error bounds can be computed by calling f07bvc, and an estimate for κ(A) can be obtained by calling f07buc with norm=Nag_InfNorm.

8 Parallelism and Performance

f07bsc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bsc 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.

9 Further Comments

The total number of real floating-point operations is approximately 8n(2kl+ku)r, assuming nkl and nku.
This function may be followed by a call to f07bvc to refine the solution and return an error estimate.
The real analogue of this function is f07bec.

10 Example

This example solves the system of equations AX=B, where
A= ( -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00+0.00i 0.00+6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00+0.00i -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00+0.00i 0.00+0.00i 4.48-1.09i -0.46-1.72i )  
and
B= ( -1.06+21.50i 12.85+02.84i -22.72-53.90i -70.22+21.57i 28.24-38.60i -20.70-31.23i -34.56+16.73i 26.01+31.97i ) .  
Here A is nonsymmetric and is treated as a band matrix, which must first be factorized by f07brc.

10.1 Program Text

Program Text (f07bsce.c)

10.2 Program Data

Program Data (f07bsce.d)

10.3 Program Results

Program Results (f07bsce.r)