NAG Library Routine Document
F07GNF (ZPPSV) computes the solution to a complex system of linear equations
Hermitian positive definite matrix stored in packed format and
||N, NRHS, LDB, INFO
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: UPLO – CHARACTER(1)Input
, the upper triangle of
If , the lower triangle of is stored.
- 2: N – INTEGERInput
On entry: , the number of linear equations, i.e., the order of the matrix .
- 3: NRHS – INTEGERInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
- 4: AP() – 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: B(LDB,) – 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: LDB – INTEGERInput
: the first dimension of the array B
as declared in the (sub)program from which F07GNF (ZPPSV) is called.
- 7: INFO – INTEGEROutput
unless the routine detects an error (see Section 6
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
If , the th argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
If , 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.
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.
9.1 Program Text
Program Text (f07gnfe.f90)
9.2 Program Data
Program Data (f07gnfe.d)
9.3 Program Results
Program Results (f07gnfe.r)