nag_real_cholesky_skyline (f01mcc) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_real_cholesky_skyline (f01mcc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

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

2  Specification

#include <nag.h>
#include <nagf01.h>
void  nag_real_cholesky_skyline (Integer n, const double a[], Integer lal, Integer row[], double al[], double d[], NagError *fail)

3  Description

nag_real_cholesky_skyline (f01mcc) determines the unit lower triangular matrix L  and the diagonal matrix D  in the Cholesky factorization A = L D LT  of a symmetric positive definite variable-bandwidth matrix A  of order n . (Such a matrix is sometimes called a ‘sky-line’ matrix.)
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 row[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 function 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 LDLT x = b  may be carried out using nag_real_cholesky_skyline_solve (f04mcc).

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  Arguments

1:     nIntegerInput
On entry: n , the order of the matrix A .
Constraint: n1 .
2:     a[lal]const doubleInput
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;
 for(i=0; i<n; ++i)
   for(j=i-row[i]+1; j<=i; ++j)
    a[k++]=matrix[i][j];
See also Section 8
3:     lalIntegerInput
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: lal row[0] + row[1] + + row[n-1] .
4:     row[n]IntegerInput
On entry: row[i]  must contain the width of row i  of the matrix A , i.e., the number of elements between the first (left-most) nonzero element and the element on the diagonal, inclusive.
Constraint: 1 row[i] i + 1 , for i=0,1,,n - 1.
5:     al[lal]doubleOutput
On exit: the elements within the envelope of the lower triangular matrix L , taken in row by row order. The envelope of L  is identical to that of the lower triangle of A . The unit diagonal elements of L  are stored explicitly. See also Section 8
6:     d[n]doubleOutput
On exit: the diagonal elements of the diagonal matrix D . Note that the determinant of A  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.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_2_INT_ARG_GT
On entry, row[value] = value while i=value . These arguments must satisfy row[i] i + 1 .
NE_2_INT_ARG_LT
On entry, lal=value  while row[0] + + row[n-1] = value. These arguments must satisfy lal row[0] + + row[n-1] .
NE_INT_ARG_LT
On entry, n=value.
Constraint: n1.
On entry, row[value]  must not be less than 1: row[value] = value.
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.

7  Accuracy

On successful exit then the computed L  and D  satisfy the relation LDLT = A + F , where
F 2 km2 ε max i a ii
and
F 2 km2 ε A 2 ,
where k  is a constant of order unity, m  is the largest value of row[i] , and ε  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 A x = b  using nag_real_cholesky_skyline_solve (f04mcc), the residual vector b - A x  should be checked.

8  Further Comments

The time taken by nag_real_cholesky_skyline (f01mcc) is approximately proportional to the sum of squares of the values of row[i] .
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 L  overwrites the lower triangle of A .

9  Example

To obtain the Cholesky factorization of the symmetric matrix, whose lower triangle is
.
For this matrix, the elements of row 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 .

9.1  Program Text

Program Text (f01mcce.c)

9.2  Program Data

Program Data (f01mcce.d)

9.3  Program Results

Program Results (f01mcce.r)


nag_real_cholesky_skyline (f01mcc) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012