NAG CL Interface
f07wsc (zpftrs)

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1 Purpose

f07wsc solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides,
AX=B ,  
using the Cholesky factorization computed by f07wrc stored in Rectangular Full Packed (RFP) format.

2 Specification

#include <nag.h>
void  f07wsc (Nag_OrderType order, Nag_RFP_Store transr, Nag_UploType uplo, Integer n, Integer nrhs, const Complex ar[], Complex b[], Integer pdb, NagError *fail)
The function may be called by the names: f07wsc, nag_lapacklin_zpftrs or nag_zpftrs.

3 Description

f07wsc is used to solve a complex Hermitian positive definite system of linear equations AX=B, the function must be preceded by a call to f07wrc which computes the Cholesky factorization of A, stored in RFP format. The RFP storage format is described in Section 3.4.3 in the F07 Chapter Introduction. The solution X is computed by forward and backward substitution.
If uplo=Nag_Upper, A=UHU, where U is upper triangular; the solution X is computed by solving UHY=B and then UX=Y.
If uplo=Nag_Lower, A=LLH, where L is lower triangular; the solution X is computed by solving LY=B and then LHX=Y.

4 References

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. Software 37, 2

5 Arguments

1: order Nag_OrderType Input
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 order=Nag_RowMajor. 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: order=Nag_RowMajor or Nag_ColMajor.
2: transr Nag_RFP_Store Input
On entry: specifies whether the normal RFP representation of A or its conjugate transpose is stored.
transr=Nag_RFP_Normal
The matrix A is stored in normal RFP format.
transr=Nag_RFP_ConjTrans
The conjugate transpose of the RFP representation of the matrix A is stored.
Constraint: transr=Nag_RFP_Normal or Nag_RFP_ConjTrans.
3: uplo Nag_UploType Input
On entry: specifies how A has been factorized.
uplo=Nag_Upper
A=UHU, where U is upper triangular.
uplo=Nag_Lower
A=LLH, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
4: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
5: nrhs Integer Input
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
6: ar[n×(n+1)/2] const Complex Input
On entry: the Cholesky factorization of A stored in RFP format, as returned by f07wrc.
7: b[dim] Complex Input/Output
Note: the dimension, dim, of the array b must be at least
  • max(1,pdb×nrhs) when order=Nag_ColMajor;
  • max(1,n×pdb) when order=Nag_RowMajor.
The (i,j)th element of the matrix B is stored in
  • b[(j-1)×pdb+i-1] when order=Nag_ColMajor;
  • b[(i-1)×pdb+j-1] when order=Nag_RowMajor.
On entry: the n×r right-hand side matrix B.
On exit: the n×r solution matrix X.
8: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax(1,n);
  • if order=Nag_RowMajor, pdbmax(1,nrhs).
9: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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 value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
NE_INT_2
On entry, pdb=value and n=value.
Constraint: pdbmax(1,n).
On entry, pdb=value and nrhs=value.
Constraint: pdbmax(1,nrhs).
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_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.

7 Accuracy

For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of equations (A+E)x=b, where c(n) is a modest linear function of n, and ε is the machine precision.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x c(n)cond(A,x)ε  
where cond(A,x)=|A-1||A||x|/xcond(A)=|A-1||A|κ(A) and κ(A) is the condition number when using the -norm.
Note that cond(A,x) can be much smaller than cond(A).

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07wsc 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.

9 Further Comments

The total number of real floating-point operations is approximately 8n2r.
The real analogue of this function is f07wec.

10 Example

This example solves the system of equations AX=B, where
A= ( 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i )  
and
B= ( 3.93-06.14i 1.48+06.58i 6.17+09.42i 4.65-04.75i -7.17-21.83i -4.91+02.29i 1.99-14.38i 7.64-10.79i ) .  
Here A is Hermitian positive definite, stored in RFP format, and must first be factorized by f07wrc.

10.1 Program Text

Program Text (f07wsce.c)

10.2 Program Data

Program Data (f07wsce.d)

10.3 Program Results

Program Results (f07wsce.r)