NAG Library Routine Document
computes the solution to a complex system of linear equations
Hermitian positive definite matrix stored in packed format and
|Integer, Intent (In)||:: ||n, nrhs, ldb|
|Integer, Intent (Out)||:: ||info|
|Complex (Kind=nag_wp), Intent (Inout)||:: ||ap(*), b(ldb,*)|
|Character (1), Intent (In)||:: ||uplo|C Header Interface
f07gnf_ (const char *uplo, const Integer *n, const Integer *nrhs, Complex ap, Complex b, const Integer *ldb, Integer *info, const Charlen length_uplo)|
The routine may be called by its
f07gnf (zppsv) uses the Cholesky decomposition to factor as if or if , where is an upper triangular matrix and is a lower triangular matrix. 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
- 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 .
On exit: if , the factor or from the Cholesky factorization or , in the same storage format as .
- 5: – 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 .
- 6: – IntegerInput
: the first dimension of the array b
as declared in the (sub)program from which f07gnf (zppsv)
- 7: – 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.
The leading minor of order of is not positive
definite, so the factorization could not be completed, and the solution has
not been 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)
for further details.
is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04cef
and returns a forward error bound and condition estimate. f04cef
calls f07gnf (zppsv)
to solve the equations.
Parallelism and Performance
f07gnf (zppsv) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07gnf (zppsv) 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 f07gaf (dppsv)
This example solves the equations
is the Hermitian positive definite matrix
Details of the Cholesky factorization of are also output.
Program Text (f07gnfe.f90)
Program Data (f07gnfe.d)
Program Results (f07gnfe.r)