NAG CL Interface
f04bgc (real_​posdef_​tridiag_​solve)

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1 Purpose

f04bgc computes the solution to a real system of linear equations AX=B, where A is an n×n symmetric positive definite tridiagonal matrix and X and B are n×r matrices. An estimate of the condition number of A and an error bound for the computed solution are also returned.

2 Specification

#include <nag.h>
void  f04bgc (Nag_OrderType order, Integer n, Integer nrhs, double d[], double e[], double b[], Integer pdb, double *rcond, double *errbnd, NagError *fail)
The function may be called by the names: f04bgc, nag_linsys_real_posdef_tridiag_solve or nag_real_sym_posdef_tridiag_lin_solve.

3 Description

A is factorized as A=LDLT, where L is a unit lower bidiagonal matrix and D is diagonal, and the factored form of A is then used to solve the system of equations.

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
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: n Integer Input
On entry: the number of linear equations n, i.e., the order of the matrix A.
Constraint: n0.
3: nrhs Integer Input
On entry: the number of right-hand sides r, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
4: d[dim] double Input/Output
Note: the dimension, dim, of the array d must be at least max(1,n).
On entry: must contain the n diagonal elements of the tridiagonal matrix A.
On exit: if fail.code= NE_NOERROR or NE_RCOND, d is overwritten by the n diagonal elements of the diagonal matrix D from the LDLT factorization of A.
5: e[dim] double Input/Output
Note: the dimension, dim, of the array e must be at least max(1,n-1).
On entry: must contain the (n-1) subdiagonal elements of the tridiagonal matrix A.
On exit: if fail.code= NE_NOERROR or NE_RCOND, e is overwritten by the (n-1) subdiagonal elements of the unit lower bidiagonal matrix L from the LDLT factorization of A. (e can also be regarded as the superdiagonal of the unit upper bidiagonal factor U from the UTDU factorization of A.)
6: 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 matrix of right-hand sides B.
On exit: if fail.code= NE_NOERROR or NE_RCOND, the n×r solution matrix X.
7: 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).
8: rcond double * Output
On exit: if fail.code= NE_NOERROR or NE_RCOND, an estimate of the reciprocal of the condition number of the matrix A, computed as rcond=1/(A1A-11).
9: errbnd double * Output
On exit: if fail.code= NE_NOERROR or NE_RCOND, an estimate of the forward error bound for a computed solution x^, such that x^-x1/x1errbnd, 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, errbnd is returned as unity.
10: 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.
The double allocatable memory required is n. In this case the factorization and the solution X have been computed, but rcond and errbnd have not been computed.
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.
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).
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_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_POS_DEF
The principal minor of order value of the matrix A is not positive definite. The factorization has not been completed and the solution could not be computed.
NE_RCOND
A solution has been computed, but rcond is less than machine precision so that the matrix A is numerically singular.

7 Accuracy

The computed solution for a single right-hand side, x^, satisfies an equation of the form
(A+E) x^=b,  
where
E1=O(ε) A1  
and ε is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κ(A) E1 A1 ,  
where κ(A)=A-11A1, the condition number of A with respect to the solution of the linear equations. f04bgc uses the approximation E1=εA1 to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f04bgc 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 total number of floating-point operations required to solve the equations AX=B is proportional to nr. The condition number estimation requires O(n) floating-point operations.
See Section 15.3 of Higham (2002) for further details on computing the condition number of tridiagonal matrices.
The complex analogue of f04bgc is f04cgc.

10 Example

This example solves the equations
AX=B,  
where A is the symmetric positive definite tridiagonal matrix
A= ( 4.0 -2.0 0 0 0 -2.0 10.0 -6.0 0 0 0 -6.0 29.0 15.0 0 0 0 15.0 25.0 8.0 0 0 0 8.0 5.0 )   and   B= ( 6.0 10.0 9.0 4.0 2.0 9.0 14.0 65.0 7.0 23.0 ) .  
An estimate of the condition number of A and an approximate error bound for the computed solutions are also printed.

10.1 Program Text

Program Text (f04bgce.c)

10.2 Program Data

Program Data (f04bgce.d)

10.3 Program Results

Program Results (f04bgce.r)