nag_herm_packed_lin_solve (f04cjc) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_herm_packed_lin_solve (f04cjc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_herm_packed_lin_solve (f04cjc) computes the solution to a complex system of linear equations AX=B, where A is an n by n complex Hermitian matrix, stored in packed format and X and B are n by r matrices. An estimate of the condition number of A and an error bound for the computed solution are also returned.

2  Specification

#include <nag.h>
#include <nagf04.h>
void  nag_herm_packed_lin_solve (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer nrhs, Complex ap[], Integer ipiv[], Complex b[], Integer pdb, double *rcond, double *errbnd, NagError *fail)

3  Description

The diagonal pivoting method is used to factor A as A=UDUH, if uplo=Nag_Upper, or A=LDLH, if uplo=Nag_Lower, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1 by 1 and 2 by 2 diagonal blocks. 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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

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 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 the matrix A is stored.
If uplo=Nag_Lower, the lower triangle of the matrix A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: the number of linear equations n, i.e., the order of the matrix A.
Constraint: n0.
4:     nrhsIntegerInput
On entry: the number of right-hand sides r, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
5:     ap[dim]ComplexInput/Output
Note: the dimension, dim, of the array ap must be at least max1,n×n+1/2.
On entry: the n by n Hermitian matrix A, packed column-wise in a linear array. The jth column of the matrix A is stored in the array ap as follows:
  • if uplo=Nag_Upper, ap[i+j-1j/2]=aij for 1ij;
  • if uplo=Nag_Lower, ap[i+j-12n-j/2]=aij for jin.
See Section 9 below for further details.
On exit: if fail.code= NE_NOERROR, the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A=UDUH or A=LDLH as computed by nag_zhptrf (f07prc), stored as a packed triangular matrix in the same storage format as A.
6:     ipiv[n]IntegerOutput
On exit: if fail.code= NE_NOERROR, details of the interchanges and the block structure of D, as determined by nag_zhptrf (f07prc).
  • If ipiv[k-1]>0, then rows and columns k and ipiv[k-1] were interchanged, and dkk is a 1 by 1 diagonal block;
  • if uplo=Nag_Upper and ipiv[k-1]=ipiv[k-2]<0, then rows and columns k-1 and -ipiv[k-1] were interchanged and dk-1:k,k-1:k is a 2 by 2 diagonal block;
  • if uplo=Nag_Lower and ipiv[k-1]=ipiv[k]<0, then rows and columns k+1 and -ipiv[k-1] were interchanged and dk:k+1,k:k+1 is a 2 by 2 diagonal block.
7:     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 matrix of right-hand sides B.
On exit: if fail.code= NE_NOERROR or NE_RCOND, the n by r solution matrix X.
8:     pdbIntegerInput
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.
9:     rconddouble *Output
On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix A, computed as rcond=1/A1A-11.
10:   errbnddouble *Output
On exit: if fail.code= NE_NOERROR or NE_RCOND, an estimate of the forward error bound for a computed solution x^, such that x^-x1/x1errbnd, where x^ is a column of the computed solution returned in the array b and x is the corresponding column of the exact solution X. If rcond is less than machine precision, then errbnd is returned as unity.
11:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
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.
A solution has been computed, but rcond is less than machine precision so that the matrix A is numerically singular.
Diagonal block value of the block diagonal matrix is zero. The factorization has been completed, but the solution could not be computed.

7  Accuracy

The computed solution for a single right-hand side, x^, satisfies an equation of the form
A+E x^=b,
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. nag_herm_packed_lin_solve (f04cjc) uses the approximation E1=εA1 to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

8  Parallelism and Performance

nag_herm_packed_lin_solve (f04cjc) is not threaded by NAG in any implementation.
nag_herm_packed_lin_solve (f04cjc) 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

The packed storage scheme is illustrated by the following example when n=4 and uplo=Nag_Upper. Two-dimensional storage of the Hermitian matrix A:
a11 a12 a13 a14 a22 a23 a24 a33 a34 a44 aij = a-ji .
Packed storage of the upper triangle of A:
ap= a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 .
The total number of floating-point operations required to solve the equations AX=B is proportional to 13n3+2n2r. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
Function nag_complex_sym_packed_lin_solve (f04djc) is for complex symmetric matrices, and the real analogue of nag_herm_packed_lin_solve (f04cjc) is nag_real_sym_packed_lin_solve (f04bjc).

10  Example

This example solves the equations
where A is the Hermitian indefinite matrix
A= -1.84i+0.00 0.11-0.11i -1.78-1.18i 3.91-1.50i 0.11+0.11i -4.63i+0.00 -1.84+0.03i 2.21+0.21i -1.78+1.18i -1.84-0.03i -8.87i+0.00 1.58-0.90i 3.91+1.50i 2.21-0.21i 1.58+0.90i -1.36i+0.00
B= 2.98-10.18i 28.68-39.89i -9.58+03.88i -24.79-08.40i -0.77-16.05i 4.23-70.02i 7.79+05.48i -35.39+18.01i .
An estimate of the condition number of A and an approximate error bound for the computed solutions are also printed.

10.1  Program Text

Program Text (f04cjce.c)

10.2  Program Data

Program Data (f04cjce.d)

10.3  Program Results

Program Results (f04cjce.r)

nag_herm_packed_lin_solve (f04cjc) (PDF version)
f04 Chapter Contents
f04 Chapter Introduction
NAG Library Manual

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