nag_dpbsvx (f07hbc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dpbsvx (f07hbc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dpbsvx (f07hbc) 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 band matrix of bandwidth 2 kd + 1  and X and B are n by r matrices. Error bounds on the solution and a condition estimate are also provided.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dpbsvx (Nag_OrderType order, Nag_FactoredFormType fact, Nag_UploType uplo, Integer n, Integer kd, Integer nrhs, double ab[], Integer pdab, double afb[], Integer pdafb, Nag_EquilibrationType *equed, double s[], double b[], Integer pdb, double x[], Integer pdx, double *rcond, double ferr[], double berr[], NagError *fail)

3  Description

nag_dpbsvx (f07hbc) performs the following steps:
  1. If fact=Nag_EquilibrateAndFactor, 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=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. If the leading i by 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. 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:     orderNag_OrderTypeInput
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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     factNag_FactoredFormTypeInput
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
afb contains the factorized form of A. If equed=Nag_Equilibrated, the matrix A has been equilibrated with scaling factors given by s. ab and afb will not be modified.
fact=Nag_NotFactored
The matrix A will be copied to afb and factorized.
fact=Nag_EquilibrateAndFactor
The matrix A will be equilibrated if necessary, then copied to afb and factorized.
Constraint: fact=Nag_Factored, Nag_NotFactored or Nag_EquilibrateAndFactor.
3:     uploNag_UploTypeInput
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:     nIntegerInput
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
5:     kdIntegerInput
On entry: kd, the number of superdiagonals of the matrix A if uplo=Nag_Upper, or the number of subdiagonals if uplo=Nag_Lower.
Constraint: kd0.
6:     nrhsIntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
7:     ab[dim]doubleInput/Output
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the upper or lower triangle of the symmetric band matrix A, except if fact=Nag_Factored and equed=Nag_Equilibrated, in which case ab must contain the equilibrated matrix DSADS.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of Aij, depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ab[kd+i-j+j-1×pdab], for j=1,,n and i=max1,j-kd,,j;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ab[i-j+j-1×pdab], for j=1,,n and i=j,,minn,j+kd;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ab[j-i+i-1×pdab], for i=1,,n and j=i,,minn,i+kd;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ab[kd+j-i+i-1×pdab], for i=1,,n and j=max1,i-kd,,i.
On exit: if fact=Nag_EquilibrateAndFactor and equed=Nag_Equilibrated, ab is overwritten by DSADS.
8:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array ab.
Constraint: pdabkd+1.
9:     afb[dim]doubleInput/Output
Note: the dimension, dim, of the array afb must be at least max1,pdafb×n.
On entry: if fact=Nag_Factored, afb contains the triangular factor U or L from the Cholesky factorization A=UTU or A=LLT of the band matrix A, in the same storage format as A. If equed=Nag_Equilibrated, afb is the factorized form of the equilibrated matrix A.
On exit: if fact=Nag_NotFactored, afb returns the triangular factor U or L from the Cholesky factorization A=UTU or A=LLT.
If fact=Nag_EquilibrateAndFactor, afb 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 ab for the form of the equilibrated matrix).
10:   pdafbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array afb.
Constraint: pdafbkd+1.
11:   equedNag_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.
12:   s[dim]doubleInput/Output
Note: the dimension, dim, of the array s must be at least max1,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.
13:   b[dim]doubleInput/Output
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×nrhs when order=Nag_ColMajor;
  • max1,n×pdb when order=Nag_RowMajor.
The i,jth 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 by 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.
14:   pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax1,n;
  • if order=Nag_RowMajor, pdbmax1,nrhs.
15:   x[dim]doubleOutput
Note: the dimension, dim, of the array x must be at least
  • max1,pdx×nrhs when order=Nag_ColMajor;
  • max1,n×pdx when order=Nag_RowMajor.
The i,jth 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 by 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.
16:   pdxIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
  • if order=Nag_ColMajor, pdxmax1,n;
  • if order=Nag_RowMajor, pdxmax1,nrhs.
17:   rconddouble *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 .
18:   ferr[nrhs]doubleOutput
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.
19:   berr[nrhs]doubleOutput
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).
20:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, kd=value.
Constraint: kd0.
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pdab=value.
Constraint: pdab>0.
On entry, pdafb=value.
Constraint: pdafb>0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdx=value.
Constraint: pdx>0.
NE_INT_2
On entry, pdab=value and kd=value.
Constraint: pdabkd+1.
On entry, pdafb=value and kd=value.
Constraint: pdafbkd+1.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
On entry, pdx=value and n=value.
Constraint: pdxmax1,n.
On entry, pdx=value and nrhs=value.
Constraint: pdxmax1,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.
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_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 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 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  Further Comments

When nk , the factorization of A  requires approximately n k+1 2  floating point operations, where k  is the number of superdiagonals.
For each right-hand side, computation of the backward error involves a minimum of 8nk  floating point operations. Each step of iterative refinement involves an additional 12nk  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 4nk  operations.
The complex analogue of this function is nag_zpbsvx (f07hpc).

9  Example

This example solves the equations
AX=B ,
where A  is the symmetric positive definite band matrix
A = 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17
and
B = 22.09 5.10 9.31 30.81 -5.24 -25.82 11.83 22.90 .
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.

9.1  Program Text

Program Text (f07hbce.c)

9.2  Program Data

Program Data (f07hbce.d)

9.3  Program Results

Program Results (f07hbce.r)


nag_dpbsvx (f07hbc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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