NAG Library Routine Document

F01MCF

The matrix A 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 NROWi of the ith 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 Ax=b. The remaining step, viz. the solution of LDLTx=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≥1.

2: A(LAL) – double precision 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 or -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, NROWi<1 or NROWi>i,
or LAL<NROW1+NROW2+…+NROWN.

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 been completed but may be very inaccurate (see Section 7).

7 Accuracy

If IFAIL=0 on exit, then the computed L and D satisfy the relation LDLT=A+F, where

F2 ≤ km2 ε × maxi aii

and

F2 ≤ km2 ε × A2 ,

where k is a constant of order unity, m is the largest value of NROWi, and ε is the machine precision. See pages 25–27 and 54–55 of Wilkinson and Reinsch (1971). If IFAIL=3 on exit, then the factorization has been completed although the matrix was not positive-definite. However the factorization may be very inaccurate and should be used only with great caution. For instance, if it is used to solve a set of equations Ax=b using F04MCF, the residual vector b-Ax should be checked.

8 Further Comments

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

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 77 and may not work in all implementations.