NAG CL Interface
f07jec (dpttrs)

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1 Purpose

f07jec 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, using the LDLT factorization returned by f07jdc.

2 Specification

#include <nag.h>
void  f07jec (Nag_OrderType order, Integer n, Integer nrhs, const double d[], const double e[], double b[], Integer pdb, NagError *fail)
The function may be called by the names: f07jec, nag_lapacklin_dpttrs or nag_dpttrs.

3 Description

f07jec should be preceded by a call to f07jdc, which computes a modified Cholesky factorization of the matrix A as
A=LDLT ,  
where L is a unit lower bidiagonal matrix and D is a diagonal matrix, with positive diagonal elements. f07jec then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form UTDU , where U is a unit upper bidiagonal matrix.

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

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: n, the order of the matrix A.
Constraint: n0.
3: nrhs Integer Input
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
4: d[dim] const double Input
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 diagonal matrix D from the LDLT factorization of A.
5: e[dim] const double Input
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 unit lower bidiagonal matrix L. (e can also be regarded as the superdiagonal of the unit upper bidiagonal matrix 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: 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: 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.
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.

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-11 A1 , the condition number of A with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Following the use of this function f07jgc can be used to estimate the condition number of A and f07jhc can be used to obtain approximate error bounds.

8 Parallelism and Performance

f07jec 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 complex analogue of this function is f07jsc.

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 0.0 0.0 -2.0 10.0 -6.0 0.0 0.0 0.0 -6.0 29.0 15.0 0.0 0.0 0.0 15.0 25.0 8.0 0.0 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 ) .  

10.1 Program Text

Program Text (f07jece.c)

10.2 Program Data

Program Data (f07jece.d)

10.3 Program Results

Program Results (f07jece.r)