Vector(1..lal, datatype=float[8]);

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NAG[f01mcc] NAG[nag_real_cholesky_skyline] - factorization of real symmetric positive-definite variable-bandwidth (skyline) matrix

Calling Sequence

f01mcc(a, row, al, d, 'n'=n, 'lal'=lal, 'fail'=fail)

nag_real_cholesky_skyline(. . .)

Parameters

a - Vector(1..lal, datatype=float[8]);

On entry: the elements within the envelope of the lower triangle of the positive-definite symmetric matrix , 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;
for i from 1 to n
   do
    for j from i-row[i]+1 to i
    do
    k:= k+1;
    a[k]=matrix[i][j]
    end do
   end do;

See also Section [Further Comments]

row - Vector(1..n, datatype=integer[kernelopts('wordsize')/8]);

On entry: must contain the width of row of the matrix , i.e., the number of elements between the first (left-most) non-zero element and the element on the diagonal, inclusive.

Constraint: , for . .

al - Vector(1..lal, datatype=float[8]);

On exit: the elements within the envelope of the lower triangular matrix , taken in row by row order. The envelope of is identical to that of the lower triangle of . The unit diagonal elements of are stored explicitly. See also Section [Further Comments]

d - Vector(1..n, datatype=float[8]);

On exit: the diagonal elements of the diagonal matrix . Note that the determinant of is equal to the product of these diagonal elements. If the value of the determinant is required it should not be determined by forming the product explicitly, because of the possibility of overflow or underflow. The logarithm of the determinant may safely be formed from the sum of the logarithms of the diagonal elements.

'n'=n - integer; (optional)

Default value: the first dimension of the arrays row, d.

On entry: , the order of the matrix .

Constraint: . .

'lal'=lal - integer; (optional)

Default value: the first dimension of the arrays a, al.

On entry: the smaller of the dimensions of the arrays a and al as declared in the function from which nag_real_cholesky_skyline (f01mcc) is called.

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_real_cholesky_skyline (f01mcc) computes the Cholesky factorization of a real symmetric positive-definite variable-bandwidth matrix.

Description

nag_real_cholesky_skyline (f01mcc) determines the unit lower triangular matrix and the diagonal matrix in the Cholesky factorization of a symmetric positive-definite variable-bandwidth matrix of order . (Such a matrix is sometimes called a "sky-line" matrix.)

The matrix is represented by the elements lying within the envelope of its lower triangular part, that is, between the first non-zero of each row and the diagonal (see Section [Example] for an example). The width of the th row is the number of elements between the first non-zero element and the element on the diagonal, inclusive. Although, of course, any matrix possesses an envelope as defined, this function is primarily intended for the factorization of symmetric positive-definite matrices with an average bandwidth which is small compared with (also see Section [Further Comments]).

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

The determination of and is normally the first of two steps in the solution of the system of equations . The remaining step, viz. the solution of may be carried out using f04mcc (nag_real_cholesky_skyline_solve).

Error Indicators and Warnings

"NE_2_INT_ARG_GT"

On entry, while . These arguments must satisfy .

"NE_2_INT_ARG_LT"

On entry, while . These arguments must satisfy .

"NE_INT_ARG_LT"

On entry, n must not be less than 1: .

"NE_NOT_POS_DEF"

The matrix is not positive-definite, possibly due to rounding errors.

"NE_NOT_POS_DEF_FACT"

The matrix is not positive-definite, possibly due to rounding errors. The factorization has been completed but may be very inaccurate.

Accuracy

On successful exit then the computed and satisfy the relation , where

and

where is a constant of order unity, is the largest value of , and is the machine precision. See pages 25–27 and 54–55 or Wilkinson and Reinsch (1971). If the error NE_NOT_POS_DEF_FACT is reported 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 using f04mcc (nag_real_cholesky_skyline_solve), the residual vector should be checked.

Further Comments

The time taken by nag_real_cholesky_skyline (f01mcc) is approximately proportional to the sum of squares of the values of .

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

The function may be called with the same actual array supplied for arguments a and al, in which case overwrites the lower triangle of .

Examples

n := 6:
lal := 14:
a := Vector([1, 2, 5, 3, 13, 16, 5, 14, 18, 8, 55, 24, 17, 77], datatype=float[8]):
arow := Vector([1, 2, 2, 1, 5, 3], datatype=integer[kernelopts('wordsize')/8]):
al := Vector(14, datatype=float[8]):
d := Vector(6, datatype=float[8]):
NAG:-f01mcc(a, arow, al, d, 'n' = n, 'lal' = lal):

al;

	See Also
	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 f01 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.