NAG Library Routine Document
computes the solution to a complex system of linear equations
Hermitian matrix stored in packed format and
|Integer, Intent (In)||:: ||n, nrhs, ldb|
|Integer, Intent (Out)||:: ||ipiv(n), info|
|Complex (Kind=nag_wp), Intent (Inout)||:: ||ap(*), b(ldb,*)|
|Character (1), Intent (In)||:: ||uplo|C Header Interface
f07pnf_ (const char *uplo, const Integer *n, const Integer *nrhs, Complex ap, Integer ipiv, Complex b, const Integer *ldb, Integer *info, const Charlen length_uplo)|
The routine may be called by its
f07pnf (zhpsv) uses the diagonal pivoting method to factor as if or if , where (or ) is a product of permutation and unit upper (lower) triangular matrices, is Hermitian and block diagonal with by and by diagonal blocks. The factored form of is then used to solve the system of equations .
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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
- 1: – Character(1)Input
, the upper triangle of
If , the lower triangle of is stored.
- 2: – IntegerInput
On entry: , the number of linear equations, i.e., the order of the matrix .
- 3: – IntegerInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
- 4: – Complex (Kind=nag_wp) arrayInput/Output
the dimension of the array ap
must be at least
, packed by columns.
- if , the upper triangle of must be stored with element in for ;
- if , the lower triangle of must be stored with element in for .
: the block diagonal matrix
and the multipliers used to obtain the factor
from the factorization
as computed by f07prf (zhptrf)
, stored as a packed triangular matrix in the same storage format as
- 5: – Integer arrayOutput
: details of the interchanges and the block structure of
. More precisely,
- if , is a by pivot block and the th row and column of were interchanged with the th row and column;
- if and , is a by pivot block and the th row and column of were interchanged with the th row and column;
- if and , is a by pivot block and the th row and column of were interchanged with the th row and column.
- 6: – Complex (Kind=nag_wp) arrayInput/Output
the second dimension of the array b
must be at least
To solve the equations
is a single right-hand side, b
may be supplied as a one-dimensional array with length
On entry: the by right-hand side matrix .
On exit: if , the by solution matrix .
- 7: – IntegerInput
: the first dimension of the array b
as declared in the (sub)program from which f07pnf (zhpsv)
- 8: – IntegerOutput
unless the routine detects an error (see Section 6
Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
Element of the diagonal is exactly zero.
The factorization has been completed, but the block diagonal matrix is exactly singular, so the solution could not be computed.
The computed solution for a single right-hand side,
, satisfies an equation of the form
is the machine precision
. An approximate error bound for the computed solution is given by
, the condition number of
with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999)
and Chapter 11 of Higham (2002)
for further details.
is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04cjf
and returns a forward error bound and condition estimate. f04cjf
calls f07pnf (zhpsv)
to solve the equations.
Parallelism and Performance
f07pnf (zhpsv) 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.
The total number of floating-point operations is approximately , where is the number of right-hand sides.
The real analogue of this routine is f07paf (dspsv)
. The complex symmetric analogue of this routine is f07qnf (zspsv)
This example solves the equations
is the Hermitian matrix
Details of the factorization of are also output.
Program Text (f07pnfe.f90)
Program Data (f07pnfe.d)
Program Results (f07pnfe.r)