nag_zpbsv (f07hnc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_zpbsv (f07hnc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zpbsv (f07hnc) computes the solution to a complex system of linear equations
AX=B ,
where A is an n by n Hermitian positive definite band matrix of bandwidth 2 kd + 1  and X and B are n by r matrices.

2  Specification

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

3  Description

nag_zpbsv (f07hnc) uses the Cholesky decomposition to factor A as A=UHU if uplo=Nag_Upper or A=LLH if uplo=Nag_Lower, where U is an upper triangular band matrix, and L is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as A. The factored form of A is then used to solve the system of equations AX=B.

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

5  Arguments

1:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     uploNag_UploTypeInput
On entry: if uplo=Nag_Upper, the upper triangle of A is stored.
If uplo=Nag_Lower, the lower triangle of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
4:     kdIntegerInput
On entry: kd, the number of superdiagonals of the matrix A if uplo=Nag_Upper, or the number of subdiagonals if uplo=Nag_Lower.
Constraint: kd0.
5:     nrhsIntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
6:     ab[dim]ComplexInput/Output
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the upper or lower triangle of the Hermitian band matrix A.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of Aij, depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ab[kd+i-j+j-1×pdab], for j=1,,n and i=max1,j-kd,,j;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ab[i-j+j-1×pdab], for j=1,,n and i=j,,minn,j+kd;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ab[j-i+i-1×pdab], for i=1,,n and j=i,,minn,i+kd;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ab[kd+j-i+i-1×pdab], for i=1,,n and j=max1,i-kd,,i.
On exit: if fail.code= NE_NOERROR, the triangular factor U or L from the Cholesky factorization A=UHU or A=LLH of the band matrix A, in the same storage format as A.
7:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A 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: if fail.code= NE_NOERROR, the n by r solution matrix X.
9:     pdbIntegerInput
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:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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.
NE_MAT_NOT_POS_DEF
The leading minor of order value of A is not positive definite, so the factorization could not be completed, and the solution has not been computed.

7  Accuracy

The computed solution for a single right-hand side, x^ , satisfies an equation of the form
A+E x^=b ,
where
E1 = Oε A1
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κA E1 A1 ,
where κA = A-11 A1 , the condition number of A  with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
nag_zpbsvx (f07hpc) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_herm_posdef_band_lin_solve (f04cfc) solves Ax=b  and returns a forward error bound and condition estimate. nag_herm_posdef_band_lin_solve (f04cfc) calls nag_zpbsv (f07hnc) to solve the equations.

8  Parallelism and Performance

nag_zpbsv (f07hnc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zpbsv (f07hnc) 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 Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

When nk , the total number of floating-point operations is approximately 4nk+12+16nkr , where k  is the number of superdiagonals and r  is the number of right-hand sides.
The real analogue of this function is nag_dpbsv (f07hac).

10  Example

This example solves the equations
Ax=b ,
where A  is the Hermitian positive definite band matrix
A = 9.39i+0.00 1.08-1.73i 0.00i+0.00 0.00i+0.00 1.08+1.73i 1.69i+0.00 -0.04+0.29i 0.00i+0.00 0.00i+0.00 -0.04-0.29i 2.65i+0.00 -0.33+2.24i 0.00i+0.00 0.00i+0.00 -0.33-2.24i 2.17i+0.00
and
b = -12.42+68.42i -9.93+00.88i -27.30-00.01i 5.31+23.63i .
Details of the Cholesky factorization of A  are also output.

10.1  Program Text

Program Text (f07hnce.c)

10.2  Program Data

Program Data (f07hnce.d)

10.3  Program Results

Program Results (f07hnce.r)


nag_zpbsv (f07hnc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014