NAG CL Interface
f07gbc (dppsvx)

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1 Purpose

f07gbc 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×n symmetric positive definite matrix stored in packed format and X and B are n×r matrices. Error bounds on the solution and a condition estimate are also provided.

2 Specification

#include <nag.h>
void  f07gbc (Nag_OrderType order, Nag_FactoredFormType fact, Nag_UploType uplo, Integer n, Integer nrhs, double ap[], double afp[], Nag_EquilibrationType *equed, double s[], double b[], Integer pdb, double x[], Integer pdx, double *rcond, double ferr[], double berr[], NagError *fail)
The function may be called by the names: f07gbc, nag_lapacklin_dppsvx or nag_dppsvx.

3 Description

f07gbc performs the following steps:
  1. 1.If fact=Nag_EquilibrateAndFactor, real diagonal scaling factors, DS , are computed to equilibrate the system:
    (DSADS) (DS-1X) = 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× DS B.
  2. 2.If fact=Nag_NotFactored or Nag_EquilibrateAndFactor, the Cholesky decomposition is used to factor the matrix A (after equilibration if fact=Nag_EquilibrateAndFactor) as A=UTU if uplo=Nag_Upper or A=LLT if uplo=Nag_Lower, where U is an upper triangular matrix and L is a lower triangular matrix.
  3. 3.If the leading i×i principal minor of A is not positive definite, then the function returns with fail.errnum=i and fail.code= NE_MAT_NOT_POS_DEF. 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, fail.code= NE_SINGULAR_WP is returned as a warning, but the function still goes on to solve for X and compute error bounds as described below.
  4. 4.The system of equations is solved for X using the factored form of A.
  5. 5.Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.
  6. 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 https://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: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: fact Nag_FactoredFormType 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=Nag_Factored
afp contains the factorized form of A. If equed=Nag_Equilibrated, the matrix A has been equilibrated with scaling factors given by s. ap and afp will not be modified.
fact=Nag_NotFactored
The matrix A will be copied to afp and factorized.
fact=Nag_EquilibrateAndFactor
The matrix A will be equilibrated if necessary, then copied to afp and factorized.
Constraint: fact=Nag_Factored, Nag_NotFactored or Nag_EquilibrateAndFactor.
3: uplo Nag_UploType Input
On entry: if uplo=Nag_Upper, the upper triangle of A is stored.
If uplo=Nag_Lower, the lower triangle of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
4: n Integer Input
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
5: nrhs Integer Input
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
6: ap[dim] double Input/Output
Note: the dimension, dim, of the array ap must be at least max(1,n×(n+1)/2).
On entry: if fact=Nag_Factored and equed=Nag_Equilibrated, ap must contain the equilibrated matrix DSADS; otherwise, ap must contain the n×n symmetric matrix A, packed by rows or columns.
The storage of elements Aij depends on the order and uplo arguments as follows:
if order=Nag_ColMajor and uplo=Nag_Upper,
Aij is stored in ap[(j-1)×j/2+i-1], for ij;
if order=Nag_ColMajor and uplo=Nag_Lower,
Aij is stored in ap[(2n-j)×(j-1)/2+i-1], for ij;
if order=Nag_RowMajor and uplo=Nag_Upper,
Aij is stored in ap[(2n-i)×(i-1)/2+j-1], for ij;
if order=Nag_RowMajor and uplo=Nag_Lower,
Aij is stored in ap[(i-1)×i/2+j-1], for ij.
On exit: if fact=Nag_Factored or Nag_NotFactored, or if fact=Nag_EquilibrateAndFactor and equed=Nag_NoEquilibration, ap is not modified.
If fact=Nag_EquilibrateAndFactor and equed=Nag_Equilibrated, ap is overwritten by DSADS.
7: afp[dim] double Input/Output
Note: the dimension, dim, of the array afp must be at least max(1,n×(n+1)/2).
On entry: if fact=Nag_Factored, afp contains the triangular factor U or L from the Cholesky factorization A=UTU or A=LLT, in the same storage format as ap. If equed=Nag_Equilibrated, afp is the factorized form of the equilibrated matrix DSADS.
On exit: if fact=Nag_NotFactored or if fact=Nag_EquilibrateAndFactor and equed=Nag_NoEquilibration, afp returns the triangular factor U or L from the Cholesky factorization A=UTU or A=LLT of the original matrix A.
If fact=Nag_EquilibrateAndFactor and equed=Nag_Equilibrated, afp 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 ap for the form of the equilibrated matrix).
8: equed Nag_EquilibrationType * Input/Output
On entry: if fact=Nag_NotFactored or Nag_EquilibrateAndFactor, equed need not be set.
If fact=Nag_Factored, equed must specify the form of the equilibration that was performed as follows:
  • if equed=Nag_NoEquilibration, no equilibration;
  • if equed=Nag_Equilibrated, equilibration was performed, i.e., A has been replaced by DSADS.
On exit: if fact=Nag_Factored, 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=Nag_Factored, equed=Nag_NoEquilibration or Nag_Equilibrated.
9: s[dim] double Input/Output
Note: the dimension, dim, of the array s must be at least max(1,n).
On entry: if fact=Nag_NotFactored or Nag_EquilibrateAndFactor, s need not be set.
If fact=Nag_Factored and equed=Nag_Equilibrated, s must contain the scale factors, DS, for A; each element of s must be positive.
On exit: if fact=Nag_Factored, s is unchanged from entry.
Otherwise, if no constraints are violated and equed=Nag_Equilibrated, s contains the scale factors, DS, for A; each element of s is positive.
10: b[dim] double Input/Output
Note: the dimension, dim, of the array b must be at least
  • max(1,pdb×nrhs) when order=Nag_ColMajor;
  • max(1,n×pdb) when order=Nag_RowMajor.
The (i,j)th element of the matrix B is stored in
  • b[(j-1)×pdb+i-1] when order=Nag_ColMajor;
  • b[(i-1)×pdb+j-1] when order=Nag_RowMajor.
On entry: the n×r right-hand side matrix B.
On exit: if equed=Nag_NoEquilibration, b is not modified.
If equed=Nag_Equilibrated, b is overwritten by DSB.
11: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax(1,n);
  • if order=Nag_RowMajor, pdbmax(1,nrhs).
12: x[dim] double Output
Note: the dimension, dim, of the array x must be at least
  • max(1,pdx×nrhs) when order=Nag_ColMajor;
  • max(1,n×pdx) when order=Nag_RowMajor.
The (i,j)th element of the matrix X is stored in
  • x[(j-1)×pdx+i-1] when order=Nag_ColMajor;
  • x[(i-1)×pdx+j-1] when order=Nag_RowMajor.
On exit: if fail.code= NE_NOERROR or NE_SINGULAR_WP, the n×r solution matrix X to the original system of equations. Note that the arrays A and B are modified on exit if equed=Nag_Equilibrated, and the solution to the equilibrated system is DS-1X.
13: pdx Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
  • if order=Nag_ColMajor, pdxmax(1,n);
  • if order=Nag_RowMajor, pdxmax(1,nrhs).
14: rcond double * 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/(A1A-11).
15: ferr[nrhs] double Output
On exit: if fail.code= NE_NOERROR or NE_SINGULAR_WP, an estimate of the forward error bound for each computed solution vector, such that x^j-xj/xjferr[j-1] 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.
16: berr[nrhs] double Output
On exit: if fail.code= NE_NOERROR or NE_SINGULAR_WP, 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).
17: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdx=value.
Constraint: pdx>0.
NE_INT_2
On entry, pdb=value and n=value.
Constraint: pdbmax(1,n).
On entry, pdb=value and nrhs=value.
Constraint: pdbmax(1,nrhs).
On entry, pdx=value and n=value.
Constraint: pdxmax(1,n).
On entry, pdx=value and nrhs=value.
Constraint: pdxmax(1,nrhs).
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MAT_NOT_POS_DEF
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.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR_WP
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+E)x=b, where c(n) 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 cond(A,x^,b) ,  
where cond(A,x^,b) = |A-1|(|A||x^|+|b|)/ x^ cond(A) = |A-1||A|κ (A) . If x^ is the j th column of X , then wc is returned in berr[j-1] and a bound on x-x^ / x^ is returned in ferr[j-1] . See Section 4.4 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

f07gbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07gbc 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 function. 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 function is f07gpc.

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 (f07gbce.c)

10.2 Program Data

Program Data (f07gbce.d)

10.3 Program Results

Program Results (f07gbce.r)