NAG CL Interface
f07hnc (zpbsv)

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1 Purpose

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

2 Specification

#include <nag.h>
void  f07hnc (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, Integer nrhs, Complex ab[], Integer pdab, Complex b[], Integer pdb, NagError *fail)
The function may be called by the names: f07hnc, nag_lapacklin_zpbsv or nag_zpbsv.

3 Description

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
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: uplo Nag_UploType Input
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: n Integer Input
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
4: kd Integer Input
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: nrhs Integer Input
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
6: ab[dim] Complex Input/Output
Note: the dimension, dim, of the array ab must be at least max(1,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=max(1,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,,min(n,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,,min(n,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=max(1,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: pdab Integer Input
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] 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: if fail.code= NE_NOERROR, the n×r solution matrix X.
9: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
  • if order=Nag_ColMajor, pdbmax(1,n);
  • if order=Nag_RowMajor, pdbmax(1,nrhs).
10: 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

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
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.
On entry, pdab=value and kd=value.
Constraint: pdabkd+1.
On entry, pdb=value and n=value.
Constraint: pdbmax(1,n).
On entry, pdb=value and nrhs=value.
Constraint: pdbmax(1,nrhs).
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.
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.
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

The computed solution for a single right-hand side, x^ , satisfies an equation of the form
(A+E) x^=b ,  
E1 = O(ε) A1  
and ε is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κ(A) E1 A1 ,  
where κ(A) = 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.
f07hpc is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04cfc solves Ax=b and returns a forward error bound and condition estimate. f04cfc calls f07hnc to solve the equations.

8 Parallelism and Performance

f07hnc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 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

When nk , the total number of floating-point operations is approximately 4n(k+1)2+16nkr , where k is the number of superdiagonals and r is the number of right-hand sides.
The real analogue of this function is 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 )  
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)