F07FBF (DPOSVX) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F07FBF (DPOSVX)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

F07FBF (DPOSVX) uses the Cholesky factorization
A=UTU   or   A=LLT  
to compute the solution to a real system of linear equations
AX=B ,  
where A is an n by n symmetric positive definite matrix and X and B are n by r matrices. Error bounds on the solution and a condition estimate are also provided.

2  Specification

SUBROUTINE F07FBF ( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
INTEGER  N, NRHS, LDA, LDAF, LDB, LDX, IWORK(N), INFO
REAL (KIND=nag_wp)  A(LDA,*), AF(LDAF,*), S(*), B(LDB,*), X(LDX,*), RCOND, FERR(NRHS), BERR(NRHS), WORK(3*N)
CHARACTER(1)  FACT, UPLO, EQUED
The routine may be called by its LAPACK name dposvx.

3  Description

F07FBF (DPOSVX) performs the following steps:
1. If FACT='E', real diagonal scaling factors, DS , are computed to equilibrate the system:
DS A DS DS-1 X = DS B .  
Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by DS A DS  and B by DS B.
2. If FACT='N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT='E') 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.
3. If the leading i by i principal minor of A is not positive definite, then the routine returns with INFO=i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO=N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.
4. The system of equations is solved for X using the factored form of A.
5. Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.
6. If equilibration was used, the matrix X is premultiplied by DS  so that it solves the original system before equilibration.

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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5  Arguments

1:     FACT – CHARACTER(1)Input
On entry: specifies whether or not the factorized form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factorized.
FACT='F'
AF contains the factorized form of A. If EQUED='Y', the matrix A has been equilibrated with scaling factors given by S. A and AF will not be modified.
FACT='N'
The matrix A will be copied to AF and factorized.
FACT='E'
The matrix A will be equilibrated if necessary, then copied to AF and factorized.
Constraint: FACT='F', 'N' or 'E'.
2:     UPLO – CHARACTER(1)Input
On entry: if UPLO='U', the upper triangle of A is stored.
If UPLO='L', the lower triangle of A is stored.
Constraint: UPLO='U' or 'L'.
3:     N – INTEGERInput
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: N0.
4:     NRHS – INTEGERInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: NRHS0.
5:     ALDA* – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,N.
On entry: the n by n symmetric matrix A.
If FACT='F' and EQUED='Y', A must have been equilibrated by the scaling factor in S as DSADS.
  • If UPLO='U', the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
  • If UPLO='L', the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: if FACT='F' or 'N', or if FACT='E' and EQUED='N', A is not modified.
If FACT='E' and EQUED='Y', A is overwritten by DSADS.
6:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07FBF (DPOSVX) is called.
Constraint: LDAmax1,N.
7:     AFLDAF* – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array AF must be at least max1,N.
On entry: if FACT='F', AF contains the triangular factor U or L from the Cholesky factorization A=UTU or A=LLT, in the same storage format as A. If EQUED'N', AF is the factorized form of the equilibrated matrix DSADS.
On exit: if FACT='N', AF returns the triangular factor U or L from the Cholesky factorization A=UTU or A=LLT of the original matrix A.
If FACT='E', AF returns the triangular factor U or L from the Cholesky factorization A=UTU or A=LLT of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix).
8:     LDAF – INTEGERInput
On entry: the first dimension of the array AF as declared in the (sub)program from which F07FBF (DPOSVX) is called.
Constraint: LDAFmax1,N.
9:     EQUED – CHARACTER(1)Input/Output
On entry: if FACT='N' or 'E', EQUED need not be set.
If FACT='F', EQUED must specify the form of the equilibration that was performed as follows:
  • if EQUED='N', no equilibration;
  • if EQUED='Y', equilibration was performed, i.e., A has been replaced by DSADS.
On exit: if FACT='F', EQUED is unchanged from entry.
Otherwise, if no constraints are violated, EQUED specifies the form of the equilibration that was performed as specified above.
Constraint: if FACT='F', EQUED='N' or 'Y'.
10:   S* – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array S must be at least max1,N.
On entry: if FACT='N' or 'E', S need not be set.
If FACT='F' and EQUED='Y', S must contain the scale factors, DS, for A; each element of S must be positive.
On exit: if FACT='F', S is unchanged from entry.
Otherwise, if no constraints are violated and EQUED='Y', S contains the scale factors, DS, for A; each element of S is positive.
11:   BLDB* – 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 right-hand side matrix B.
On exit: if EQUED='N', B is not modified.
If EQUED='Y', B is overwritten by DSB.
12:   LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07FBF (DPOSVX) is called.
Constraint: LDBmax1,N.
13:   XLDX* – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array X must be at least max1,NRHS.
On exit: if INFO=0 or N+1, the n by r solution matrix X to the original system of equations. Note that the arrays A and B are modified on exit if EQUED='Y', and the solution to the equilibrated system is DS-1X.
14:   LDX – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which F07FBF (DPOSVX) is called.
Constraint: LDXmax1,N.
15:   RCOND – REAL (KIND=nag_wp)Output
On exit: if no constraints are violated, an estimate of the reciprocal condition number of the matrix A (after equilibration if that is performed), computed as RCOND=1.0/A1 A-11 .
16:   FERRNRHS – REAL (KIND=nag_wp) arrayOutput
On exit: if INFO=0 or N+1, an estimate of the forward error bound for each computed solution vector, such that x^j-xj/xjFERRj where x^j is the jth column of the computed solution returned in the array X and xj is the corresponding column of the exact solution X. The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.
17:   BERRNRHS – REAL (KIND=nag_wp) arrayOutput
On exit: if INFO=0 or N+1, an estimate of the component-wise relative backward error of each computed solution vector x^j (i.e., the smallest relative change in any element of A or B that makes x^j an exact solution).
18:   WORK3×N – REAL (KIND=nag_wp) arrayWorkspace
19:   IWORKN – INTEGER arrayWorkspace
20:   INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
INFO>0andINFON
The leading minor of order value of A is not positive definite, so the factorization could not be completed, and the solution has not been computed. RCOND=0.0 is returned.
INFO=N+1
U (or L) is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest.

7  Accuracy

For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of equations A+Ex=b, where cn is a modest linear function of n, and ε is the machine precision. See Section 10.1 of Higham (2002) for further details.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x^ wc condA,x^,b  
where condA,x^,b = A-1 A x^ + b / x^ condA = A-1 A κ A. If x^  is the j th column of X , then wc  is returned in BERRj  and a bound on x - x^ / x^  is returned in FERRj . See Section 4.4 of Anderson et al. (1999) for further details.

8  Parallelism and Performance

F07FBF (DPOSVX) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F07FBF (DPOSVX) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The factorization of A  requires approximately 13 n3  floating-point operations.
For each right-hand side, computation of the backward error involves a minimum of 4n2  floating-point operations. Each step of iterative refinement involves an additional 6n2  operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form Ax=b ; the number is usually 4 or 5 and never more than 11. Each solution involves approximately 2n2  operations.
The complex analogue of this routine is F07FPF (ZPOSVX).

10  Example

This example solves the equations
AX=B ,  
where A  is the symmetric positive definite matrix
A = 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18  
and
B = 8.70 8.30 -13.35 2.13 1.89 1.61 -4.14 5.00 .  
Error estimates for the solutions, information on equilibration and an estimate of the reciprocal of the condition number of the scaled matrix A  are also output.

10.1  Program Text

Program Text (f07fbfe.f90)

10.2  Program Data

Program Data (f07fbfe.d)

10.3  Program Results

Program Results (f07fbfe.r)


F07FBF (DPOSVX) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016