NAG Library Function Document

nag_dpftrs (f07wec)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_dpftrs (f07wec) solves a real symmetric positive definite system of linear equations with multiple right-hand sides,
AX=B ,  
using the Cholesky factorization computed by nag_dpftrf (f07wdc) stored in Rectangular Full Packed (RFP) format.

2
Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dpftrs (Nag_OrderType order, Nag_RFP_Store transr, Nag_UploType uplo, Integer n, Integer nrhs, const double ar[], double b[], Integer pdb, NagError *fail)

3
Description

nag_dpftrs (f07wec) is used to solve a real symmetric positive definite system of linear equations AX=B, the function must be preceded by a call to nag_dpftrf (f07wdc) which computes the Cholesky factorization of A, stored in RFP format. The RFP storage format is described in Section 3.3.3 in the f07 Chapter Introduction. The solution X is computed by forward and backward substitution.
If uplo=Nag_Upper, A=UTU, where U is upper triangular; the solution X is computed by solving UTY=B and then UX=Y.
If uplo=Nag_Lower, A=LLT, where L is lower triangular; the solution X is computed by solving LY=B and then LTX=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_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 order=Nag_RowMajor. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     transr Nag_RFP_StoreInput
On entry: specifies whether the RFP representation of A is normal or transposed.
transr=Nag_RFP_Normal
The matrix A is stored in normal RFP format.
transr=Nag_RFP_Trans
The matrix A is stored in transposed RFP format.
Constraint: transr=Nag_RFP_Normal or Nag_RFP_Trans.
3:     uplo Nag_UploTypeInput
On entry: specifies how A has been factorized.
uplo=Nag_Upper
A=UTU, where U is upper triangular.
uplo=Nag_Lower
A=LLT, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
4:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
5:     nrhs IntegerInput
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
6:     ar[n×n+1/2] const doubleInput
On entry: the Cholesky factorization of A stored in RFP format, as returned by nag_dpftrf (f07wdc).
7:     b[dim] doubleInput/Output
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×nrhs when order=Nag_ColMajor;
  • max1,n×pdb when order=Nag_RowMajor.
The i,jth 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 by r right-hand side matrix B.
On exit: the n by r solution matrix X.
8:     pdb IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax1,n;
  • if order=Nag_RowMajor, pdbmax1,nrhs.
9:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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+Ex=b, where cn 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 cncondA,xε  
where condA,x=A-1Ax/xcondA=A-1AκA and κA is the condition number when using the -norm.
Note that condA,x can be much smaller than condA.

8
Parallelism and Performance

nag_dpftrs (f07wec) 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 floating-point operations is approximately 2n2r.
The complex analogue of this function is nag_zpftrs (f07wsc).

10
Example

This example solves the system of equations AX=B, where
A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18   and   B= 8.70 8.30 -13.35 2.13 1.89 1.61 -4.14 5.00 .  
Here A is symmetric positive definite, stored in RFP format, and must first be factorized by nag_dpftrf (f07wdc).

10.1
Program Text

Program Text (f07wece.c)

10.2
Program Data

Program Data (f07wece.d)

10.3
Program Results

Program Results (f07wece.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017