NAG C Library Function Document

nag_zpbtrs (f07hsc)

1
Purpose

nag_zpbtrs (f07hsc) solves a complex Hermitian positive definite band system of linear equations with multiple right-hand sides,
AX=B ,  
where A has been factorized by nag_zpbtrf (f07hrc).

2
Specification

#include <nag.h>
#include <nagf07.h>
void  nag_zpbtrs (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, Integer nrhs, const Complex ab[], Integer pdab, Complex b[], Integer pdb, NagError *fail)

3
Description

nag_zpbtrs (f07hsc) is used to solve a complex Hermitian positive definite band system of linear equations AX=B, the function must be preceded by a call to nag_zpbtrf (f07hrc) which computes the Cholesky factorization of A. The solution X is computed by forward and backward substitution.
If uplo=Nag_Upper, A=UHU, where U is upper triangular; the solution X is computed by solving UHY=B and then UX=Y.
If uplo=Nag_Lower, A=LLH, where L is lower triangular; the solution X is computed by solving LY=B and then LHX=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_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 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: order=Nag_RowMajor or Nag_ColMajor.
2:     uplo Nag_UploTypeInput
On entry: specifies how A has been factorized.
uplo=Nag_Upper
A=UHU, where U is upper triangular.
uplo=Nag_Lower
A=LLH, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     kd IntegerInput
On entry: kd, the number of superdiagonals or subdiagonals of the matrix A.
Constraint: kd0.
5:     nrhs IntegerInput
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
6:     ab[dim] const ComplexInput
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the Cholesky factor of A, as returned by nag_zpbtrf (f07hrc).
7:     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: pdabkd+1.
8:     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.
9:     pdb IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax1,n;
  • if order=Nag_RowMajor, pdbmax1,nrhs.
10:   fail 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 value had an illegal value.
NE_INT
On entry, kd=value.
Constraint: kd0.
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, pdab=value and kd=value.
Constraint: pdabkd+1.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
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.

7
Accuracy

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 ck+1 is a modest linear function of k+1, and ε is the machine precision.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x ck+1condA,xε  
where condA,x=A-1Ax/xcondA=A-1AκA. Note that condA,x can be much smaller than condA.
Forward and backward error bounds can be computed by calling nag_zpbrfs (f07hvc), and an estimate for κA (=κ1A) can be obtained by calling nag_zpbcon (f07huc).

8
Parallelism and Performance

nag_zpbtrs (f07hsc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zpbtrs (f07hsc) 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 16nkr, assuming nk.
This function may be followed by a call to nag_zpbrfs (f07hvc) to refine the solution and return an error estimate.
The real analogue of this function is nag_dpbtrs (f07hec).

10
Example

This example solves the system of equations AX=B, where
A= 9.39+0.00i 1.08-1.73i 0.00+0.00i 0.00+0.00i 1.08+1.73i 1.69+0.00i -0.04+0.29i 0.00+0.00i 0.00+0.00i -0.04-0.29i 2.65+0.00i -0.33+2.24i 0.00+0.00i 0.00+0.00i -0.33-2.24i 2.17+0.00i  
and
B= -12.42+68.42i 54.30-56.56i -9.93+00.88i 18.32+04.76i -27.30-00.01i -4.40+09.97i 5.31+23.63i 9.43+01.41i .  
Here A is Hermitian positive definite, and is treated as a band matrix, which must first be factorized by nag_zpbtrf (f07hrc).

10.1
Program Text

Program Text (f07hsce.c)

10.2
Program Data

Program Data (f07hsce.d)

10.3
Program Results

Program Results (f07hsce.r)