NAG FL Interface
f07mnf (zhesv)

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1 Purpose

f07mnf computes the solution to a complex system of linear equations
AX=B ,  
where A is an n×n Hermitian matrix and X and B are n×r matrices.

2 Specification

Fortran Interface
Subroutine f07mnf ( uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info)
Integer, Intent (In) :: n, nrhs, lda, ldb, lwork
Integer, Intent (Inout) :: ipiv(*)
Integer, Intent (Out) :: info
Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*)
Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
Character (1), Intent (In) :: uplo
C Header Interface
#include <nag.h>
void  f07mnf_ (const char *uplo, const Integer *n, const Integer *nrhs, Complex a[], const Integer *lda, Integer ipiv[], Complex b[], const Integer *ldb, Complex work[], const Integer *lwork, Integer *info, const Charlen length_uplo)
The routine may be called by the names f07mnf, nagf_lapacklin_zhesv or its LAPACK name zhesv.

3 Description

f07mnf uses the diagonal pivoting method to factor A as A=UDUH if uplo='U' or A=LDLH if uplo='L', where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1×1 and 2×2 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 https://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: uplo Character(1) Input
On entry: if uplo='U', the upper triangle of A is stored.
If uplo='L', the lower triangle of A is stored.
Constraint: uplo='U' or 'L'.
2: n Integer Input
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
3: nrhs Integer Input
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
4: a(lda,*) Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,n).
On entry: the n×n Hermitian matrix A.
  • If uplo='U', the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
  • If uplo='L', the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: if info=0, 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 f07mrf.
5: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f07mnf is called.
Constraint: ldamax(1,n).
6: ipiv(*) Integer array Output
Note: the dimension of the array ipiv must be at least max(1,n).
On exit: details of the interchanges and the block structure of D. More precisely,
  • if ipiv(i)=k>0, dii is a 1×1 pivot block and the ith row and column of A were interchanged with the kth row and column;
  • if uplo='U' and ipiv(i-1)=ipiv(i)=-l<0, (di-1,i-1d¯i,i-1 d¯i,i-1dii ) is a 2×2 pivot block and the (i-1)th row and column of A were interchanged with the lth row and column;
  • if uplo='L' and ipiv(i)=ipiv(i+1)=-m<0, (diidi+1,idi+1,idi+1,i+1) is a 2×2 pivot block and the (i+1)th row and column of A were interchanged with the mth row and column.
7: b(ldb,*) Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least max(1,nrhs).
to solve the equations Ax=b, where b is a single right-hand side, b may be supplied as a one-dimensional array with length ldb=max(1,n).
On entry: the n×r right-hand side matrix B.
On exit: if info=0, the n×r solution matrix X.
8: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07mnf is called.
Constraint: ldbmax(1,n).
9: work(max(1,lwork)) Complex (Kind=nag_wp) array Workspace
On exit: if info=0, work(1) returns the optimal lwork.
10: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f07mnf is called.
lwork1, and for best performance lworkmax(1,n×nb), where nb is the optimal block size for f07mrf.
If lwork=−1, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
11: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
info>0
Element value of the diagonal is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so 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 ,  
where
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.
f07mpf is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04chf solves Ax=b and returns a forward error bound and condition estimate. f04chf calls f07mnf to solve the equations.

8 Parallelism and Performance

f07mnf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of floating-point operations is approximately 43 n3 + 8n2r , where r is the number of right-hand sides.
The real analogue of this routine is f07maf. The complex symmetric analogue of this routine is f07nnf.

10 Example

This example solves the equations
Ax=b ,  
where A is the Hermitian 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 )  
and
b = ( 2.98-10.18i -9.58+03.88i -0.77-16.05i 7.79+05.48i ) .  
Details of the factorization of A are also output.

10.1 Program Text

Program Text (f07mnfe.f90)

10.2 Program Data

Program Data (f07mnfe.d)

10.3 Program Results

Program Results (f07mnfe.r)