NAG C Library Function Document

nag_zgbtrs (f07bsc)


nag_zgbtrs (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 nag_zgbtrf (f07brc).


#include <nag.h>
#include <nagf07.h>
void  nag_zgbtrs (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)


nag_zgbtrs (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 nag_zgbtrf (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.


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


1:     order 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 order=Nag_RowMajor. See Section in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     trans Nag_TransTypeInput
On entry: indicates the form of the equations.
AX=B is solved for X.
ATX=B is solved for X.
AHX=B is solved for X.
Constraint: trans=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
3:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     kl IntegerInput
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
5:     ku IntegerInput
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
6:     nrhs IntegerInput
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
7:     ab[dim] const ComplexInput
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the LU factorization of A, as returned by nag_zgbtrf (f07brc).
8:     pdab IntegerInput
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 IntegerInput
Note: the dimension, dim, of the array ipiv must be at least max1,n.
On entry: the pivot indices, as returned by nag_zgbtrf (f07brc).
10:   b[dim] ComplexInput/Output
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×nrhs when order=Nag_ColMajor;
  • max1,n×pdb when order=Nag_RowMajor.
The i,jth 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 by r right-hand side matrix B.
On exit: the n by r solution matrix X.
11:   pdb IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
  • if order=Nag_ColMajor, pdbmax1,n;
  • if order=Nag_RowMajor, pdbmax1,nrhs.
12:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
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.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
On entry, pdab=value, kl=value and ku=value.
Constraint: pdab2×kl+ku+1.
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.
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.


For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of equations A+Ex=b, where
EckεLU ,  
ck 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 ckcondA,xε  
where condA,x=A-1Ax/xcondA=A-1AκA.
Note that condA,x can be much smaller than condA, and condAH (which is the same as condAT) can be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling nag_zgbrfs (f07bvc), and an estimate for κA can be obtained by calling nag_zgbcon (f07buc) with norm=Nag_InfNorm.

Parallelism and Performance

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

Further Comments

The total number of real floating-point operations is approximately 8n2kl+kur, assuming nkl and nku.
This function may be followed by a call to nag_zgbrfs (f07bvc) to refine the solution and return an error estimate.
The real analogue of this function is nag_dgbtrs (f07bec).


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  
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 nag_zgbtrf (f07brc).

Program Text

Program Text (f07bsce.c)

Program Data

Program Data (f07bsce.d)

Program Results

Program Results (f07bsce.r)