The routine may be called by the names f07gaf, nagf_lapacklin_dppsv or its LAPACK name dppsv.
f07gaf 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 https://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
On entry: if , the upper triangle of is stored.
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: – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array ap
must be at least
On entry: the symmetric matrix , 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: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b
must be at least
On entry: the right-hand side matrix .
On exit: if , the solution matrix .
6: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07gaf is called.
7: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error 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
and is the machine precision. An approximate error bound for the computed solution is given by
where , 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.
f07gbf is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04bef solves and returns a forward error bound and condition estimate. f04bef calls f07gaf to solve the equations.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07gaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07gaf 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.