NAG Library Routine Document

F01MCF

is represented by the elements lying within the envelope of its lower triangular part, that is, between the first nonzero of each row and the diagonal (see Section 9 for an example). The width

NROW (i)

of the

i

th row is the number of elements between the first nonzero element and the element on the diagonal, inclusive. Although, of course, any matrix possesses an envelope as defined, this routine is primarily intended for the factorization of symmetric positive definite matrices with an average bandwidth which is small compared with

n

(also see Section 8).

The method is based on the property that during Cholesky factorization there is no fill-in outside the envelope.

The determination of

L

and

D

is normally the first of two steps in the solution of the system of equations

A x = b

. The remaining step, viz. the solution of

L D L^{T} x = b

, may be carried out using F04MCF.

4 References

Jennings A (1966) A compact storage scheme for the solution of symmetric linear simultaneous equations Comput. J. 9 281–285

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 \geq 1

2: A(LAL) – REAL (KIND=nag_wp) arrayInput

On entry: the elements within the envelope of the lower triangle of the positive definite symmetric matrix

A

, taken in row by row order. The following code assigns the matrix elements within the envelope to the correct elements of the array:

    K = 0
    DO 20 I = 1, N
       DO 10 J = I-NROW(I)+1, I
          K = K + 1
          A(K) = matrix (I,J)
 10    CONTINUE
 20 CONTINUE

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$N < 1$ ,
or	for some $i$ , $NROW (i) < 1$ or $NROW (i) > i$ ,
or	$LAL < NROW (1) + NROW (2) + \dots + NROW (N)$ .

$IFAIL = 2$: $A$ is not positive definite, or this property has been destroyed by rounding errors. The factorization has not been completed.

$IFAIL = 3$: $A$ is not positive definite, or this property has been destroyed by rounding errors. The factorization has not been completed.

7 Accuracy

IFAIL = 0

on exit, then the computed

L

and

D

satisfy the relation

L D L^{T} = A + F

, where

{‖F‖}_{2} \leq k m^{2} ε \times \max_{i} a_{i i}

and

{‖F‖}_{2} \leq k m^{2} ε \times {‖A‖}_{2},

where

k

is a constant of order unity,

m

is the largest value of

NROW (i)

, and

ε

is the machine precision. See pages 25–27 and 54–55 of Wilkinson and Reinsch (1971).

8 Further Comments

The time taken by F01MCF is approximately proportional to the sum of squares of the values of

NROW (i)

The distribution of row widths may be very non-uniform without undue loss of efficiency. Moreover, the routine has been designed to be as competitive as possible in speed with routines designed for full or uniformly banded matrices, when applied to such matrices.

Unless otherwise stated in the Users' Note for your implementation, the routine may be called with the same actual array supplied for parameters A and AL, in which case

L

overwrites the lower triangle of A. However this is not standard Fortran and may not work in all implementations.

9 Example

This example obtains the Cholesky factorization of the symmetric matrix, whose lower triangle is:

(\begin{array}{r} 1 \\ 2 & 5 \\ 0 & 3 & 13 \\ 0 & 0 & 0 & 16 \\ 5 & 14 & 18 & 8 & 55 \\ 0 & 0 & 0 & 24 & 17 & 77 \end{array}) .

For this matrix, the elements of NROW must be set to

1

2

2

1

5

3

, and the elements within the envelope must be supplied in row order as:

1, 2, 5, 3, 13, 16, 5, 14, 18, 8, 55, 24, 17, 77 .

NAG Library Routine DocumentF01MCF

+− Contents

1 Purpose

2 Specification

3 Description