F04AGF (PDF version)
F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F04AGF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+− Contents

    1  Purpose
    2  Specification
    3  Description
    4  References
    5  Parameters
    6  Error Indicators and Warnings
    7  Accuracy
    8  Further Comments
+− 9  Example

1  Purpose

F04AGF calculates the approximate solution of a set of real symmetric positive definite linear equations with multiple right-hand sides, AX=B, where A has been factorized by F03AEF.

2  Specification

SUBROUTINE F04AGF ( N, IR, A, LDA, P, B, LDB, X, LDX)
INTEGER  N, IR, LDA, LDB, LDX
REAL (KIND=nag_wp)  A(LDA,N), P(N), B(LDB,IR), X(LDX,IR)

3  Description

To solve a set of real linear equations AX=B where A is symmetric positive definite, F04AGF must be preceded by a call to F03AEF which computes a Cholesky factorization of A as A=LLT, where L is lower triangular. The columns x of the solution X are found by forward and backward substitution in Ly=b and LTx=y, where b is a column of the right-hand sides.

4  References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

5  Parameters

1:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N≥1.
2:     IR – INTEGERInput
On entry: r, the number of right-hand sides.
3:     A(LDA,N) – REAL (KIND=nag_wp) arrayInput
On entry: the upper triangle of the n by n positive definite symmetric matrix A, and the subdiagonal elements of its Cholesky factor L, as returned by F03AEF.
On exit: is used as internal workspace, but is restored on exit.
4:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F04AGF is called.
Constraint: LDA≥N.
5:     P(N) – REAL (KIND=nag_wp) arrayInput
On entry: the reciprocals of the diagonal elements of L, as returned by F03AEF.
On exit: is used as internal workspace, but is restored on exit.
6:     B(LDB,IR) – REAL (KIND=nag_wp) arrayInput
On entry: the n by r right-hand side matrix B. See also Section 8.
7:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F04AGF is called.
Constraint: LDB≥N.
8:     X(LDX,IR) – REAL (KIND=nag_wp) arrayOutput
On exit: the n by r solution matrix X. See also Section 8.
9:     LDX – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which F04AGF is called.
Constraint: LDX≥N.

6  Error Indicators and Warnings

If an error is detected in an input parameter F04AGF will act as if a soft noisy exit has been requested (see Section 3.3.4 in the Essential Introduction).

7  Accuracy

The accuracy of the computed solutions depends on the conditioning of the original matrix. For a detailed error analysis see page 39 of Wilkinson and Reinsch (1971).

8  Further Comments

The time taken is approximately proportional to n2r.

9  Example

This example solves the set of linear equations AX=B where
A= 5 7 6 5 7 10 8 7 6 8 10 9 5 7 9 10   and  B= 23 32 33 31 .

9.1  Program Text

Program Text (f04agfe.f90)

9.2  Program Data

Program Data (f04agfe.d)

9.3  Program Results

Program Results (f04agfe.r)

F04AGF (PDF version)
F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual
© The Numerical Algorithms Group Ltd, Oxford, UK. 2012