NAG Library Routine Document
F04BEF
1 Purpose
F04BEF computes the solution to a real system of linear equations AX=B, where A is an n by n symmetric positive definite matrix, stored in packed format, and X and B are n by r matrices. An estimate of the condition number of A and an error bound for the computed solution are also returned.
2 Specification
INTEGER |
N, NRHS, LDB, IFAIL |
REAL (KIND=nag_wp) |
AP(*), B(LDB,*), RCOND, ERRBND |
CHARACTER(1) |
UPLO |
|
3 Description
The Cholesky factorization is used to factor A as A=UTU, if UPLO='U', or A=LLT, if UPLO='L', where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of A is then used to solve the system of equations AX=B.
4 References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lug
Higham N J (2002)
Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5 Parameters
- 1: UPLO – CHARACTER(1)Input
On entry: if
UPLO='U', the upper triangle of the matrix
A is stored.
If UPLO='L', the lower triangle of the matrix A is stored.
Constraint:
UPLO='U' or 'L'.
- 2: N – INTEGERInput
On entry: the number of linear equations n, i.e., the order of the matrix A.
Constraint:
N≥0.
- 3: NRHS – INTEGERInput
On entry: the number of right-hand sides r, i.e., the number of columns of the matrix B.
Constraint:
NRHS≥0.
- 4: AP(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
AP
must be at least
max1,N×N+1/2.
On entry: the
n by
n symmetric matrix
A. The upper or lower triangular part of the symmetric matrix is packed column-wise in a linear array. The
jth column of
A is stored in the array
AP as follows:
- if UPLO='U', APi+j-1j/2=aij for 1≤i≤j;
- if UPLO='L', APi+j-12n-j/2=aij for j≤i≤n.
See
Section 8 below for further details.
On exit: if IFAIL=0 or N+1, the factor U or L from the Cholesky factorization A=UTU or A=LLT, in the same storage format as A.
- 5: B(LDB,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
B
must be at least
max1,NRHS.
On entry: the n by r matrix of right-hand sides B.
On exit: if IFAIL=0 or N+1, the n by r solution matrix X.
- 6: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F04BEF is called.
Constraint:
LDB≥max1,N.
- 7: RCOND – REAL (KIND=nag_wp)Output
On exit: if IFAIL=0 or N+1, an estimate of the reciprocal of the condition number of the matrix A, computed as RCOND=1/A1A-11.
- 8: ERRBND – REAL (KIND=nag_wp)Output
On exit: if
IFAIL=0 or
N+1, an estimate of the forward error bound for a computed solution
x^, such that
x^-x1/x1≤ERRBND, where
x^ is a column of the computed solution returned in the array
B and
x is the corresponding column of the exact solution
X. If
RCOND is less than
machine precision, then
ERRBND is returned as unity.
- 9: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
0.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
On exit:
IFAIL=0 unless the routine detects an error or a warning has been flagged (see
Section 6).
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<0 and IFAIL≠-999
If IFAIL=-i, the ith argument had an illegal value.
- IFAIL=-999
Allocation of memory failed. The integer allocatable memory required is
N, and the real allocatable memory required is
3×N. Allocation failed before the solution could be computed.
- IFAIL>0 and IFAIL≤N
If IFAIL=i, the leading minor of order i of A is not positive definite. The factorization could not be completed, and the solution has not been computed.
- IFAIL=N+1
RCOND is less than
machine precision, so that the matrix
A is numerically singular. A solution to the equations
AX=B has nevertheless been computed.
7 Accuracy
The computed solution for a single right-hand side,
x^, satisfies an equation of the form
where
and
ε is the
machine precision. An approximate error bound for the computed solution is given by
where
κA=A-11A1, the condition number of
A with respect to the solution of the linear equations. F04BEF uses the approximation
E1=εA1 to estimate
ERRBND. See Section 4.4 of
Anderson et al. (1999) for further details.
8 Further Comments
The packed storage scheme is illustrated by the following example when
n=4 and
UPLO='U'. Two-dimensional storage of the symmetric matrix
A:
Packed storage of the upper triangle of
A:
The total number of floating point operations required to solve the equations AX=B is proportional to 13n3+n2r. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of
Higham (2002) for further details.
The complex analogue of F04BEF is
F04CEF.
9 Example
This example solves the equations
where
A is the symmetric positive definite matrix
An estimate of the condition number of A and an approximate error bound for the computed solutions are also printed.
9.1 Program Text
Program Text (f04befe.f90)
9.2 Program Data
Program Data (f04befe.d)
9.3 Program Results
Program Results (f04befe.r)