The function may be called by the names: f07gac, nag_lapacklin_dppsv or nag_dppsv.
3Description
f07gac 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 .
4References
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
5Arguments
1: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
2: – Nag_UploTypeInput
On entry: if , the upper triangle of is stored.
If , the lower triangle of is stored.
Constraint:
or .
3: – IntegerInput
On entry: , the number of linear equations, i.e., the order of the matrix .
Constraint:
.
4: – IntegerInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
Constraint:
.
5: – doubleInput/Output
Note: the dimension, dim, of the array ap
must be at least
.
On entry: the symmetric matrix , packed by rows or columns.
The storage of elements depends on the order and uplo arguments as follows:
if and ,
is stored in , for ;
if and ,
is stored in , for ;
if and ,
is stored in , for ;
if and ,
is stored in , for .
On exit: if NE_NOERROR, the factor or from the Cholesky factorization or , in the same storage format as .
6: – doubleInput/Output
Note: the dimension, dim, of the array b
must be at least
when
;
when
.
The th element of the matrix is stored in
when ;
when .
On entry: the right-hand side matrix .
On exit: if NE_NOERROR, the solution matrix .
7: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
if ,
;
if , .
8: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_INT
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, . Constraint: .
NE_INT_2
On entry, and .
Constraint: .
On entry, and .
Constraint: .
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MAT_NOT_POS_DEF
The leading minor of order of is not positive definite, so the factorization could not be completed, and the solution has not been computed.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
7Accuracy
The computed solution for a single right-hand side, , satisfies an equation of the form
where
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.
f07gbc is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04bec solves and returns a forward error bound and condition estimate. f04bec calls f07gac to solve the equations.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07gac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07gac 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The total number of floating-point operations is approximately , where is the number of right-hand sides.