f07wjc computes the inverse of a real symmetric positive definite matrix using the Cholesky factorization computed by f07wdc stored in Rectangular Full Packed (RFP) format.
The function may be called by the names: f07wjc, nag_lapacklin_dpftri or nag_dpftri.
3Description
f07wjc is used to compute the inverse of a real symmetric positive definite matrix , stored in RFP format.
The RFP storage format is described in Section 3.4.3 in the F07 Chapter Introduction.
The function must be preceded by a call to f07wdc, which computes the Cholesky factorization of .
If , and is computed by first inverting and then forming .
If , and is computed by first inverting and then forming .
4References
Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal.12 1–19
Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software37, 2
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_RFP_StoreInput
On entry: specifies whether the RFP representation of is normal or transposed.
The matrix is stored in normal RFP format.
The matrix is stored in transposed RFP format.
Constraint:
or .
3: – Nag_UploTypeInput
On entry: specifies how has been factorized.
, where is upper triangular.
, where is lower triangular.
Constraint:
or .
4: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
5: – doubleInput/Output
On entry: the Cholesky factorization of stored in RFP format, as returned by f07wdc.
On exit: the factorization is overwritten by the matrix stored in RFP format.
6: – 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: .
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 is not positive definite and the factorization could not be completed. Hence itself is not positive definite. This may indicate an error in forming the matrix . There is no function specifically designed to invert a symmetric matrix stored in RFP format which is not positive definite; the matrix must be treated as a full symmetric matrix, by calling f07mjc.
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 inverse satisfies
where is a modest function of , is the machine precision and is the condition number of defined by
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07wjc 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 .