NAG CL Interface
f04ccc (complex_​tridiag_​solve)

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1 Purpose

f04ccc computes the solution to a complex system of linear equations AX=B, where A is an n×n 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  f04ccc (Nag_OrderType order, Integer n, Integer nrhs, Complex dl[], Complex d[], Complex du[], Complex du2[], Integer ipiv[], Complex b[], Integer pdb, double *rcond, double *errbnd, NagError *fail)
The function may be called by the names: f04ccc, nag_linsys_complex_tridiag_solve or nag_complex_tridiag_lin_solve.

3 Description

The LU decomposition with partial pivoting and row interchanges is used to factor A as A=PLU, where P is a permutation matrix, L is unit lower triangular with at most one nonzero subdiagonal element, and U is an upper triangular band matrix with two superdiagonals. The factored form of A is then used to solve the system of equations AX=B.
Note that the equations ATX=B may be solved by interchanging the order of the arguments du and dl.

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: dl[dim] Complex Input/Output
Note: the dimension, dim, of the array dl must be at least max(1,n-1).
On entry: must contain the (n-1) subdiagonal elements of the matrix A.
On exit: if fail.code= NE_NOERROR, dl is overwritten by the (n-1) multipliers that define the matrix L from the LU factorization of A.
5: d[dim] Complex 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 matrix A.
On exit: if fail.code= NE_NOERROR, d is overwritten by the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
6: du[dim] Complex Input/Output
Note: the dimension, dim, of the array du must be at least max(1,n-1).
On entry: must contain the (n-1) superdiagonal elements of the matrix A
On exit: if fail.code= NE_NOERROR, du is overwritten by the (n-1) elements of the first superdiagonal of U.
7: du2[n-2] Complex Output
On exit: if fail.code= NE_NOERROR, du2 returns the (n-2) elements of the second superdiagonal of U.
8: ipiv[n] Integer Output
On exit: if fail.code= NE_NOERROR, the pivot indices that define the permutation matrix P; at the ith step row i of the matrix was interchanged with row ipiv[i-1]. ipiv[i-1] will always be either i or (i+1); ipiv[i-1]=i indicates a row interchange was not required.
9: b[dim] Complex 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.
10: 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).
11: rcond double * Output
On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix A, computed as rcond=1/(A1A-11).
12: 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.
13: 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 Complex allocatable memory required is 2×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_RCOND
A solution has been computed, but rcond is less than machine precision so that the matrix A is numerically singular.
NE_SINGULAR
Diagonal element value of the upper triangular factor is zero. The factorization has been completed, but the solution could not be computed.

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. f04ccc uses the approximation E1=εA1 to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

f04ccc 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 typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The real analogue of f04ccc is f04bcc.

10 Example

This example solves the equations
AX=B,  
where A is the tridiagonal matrix
A= ( -1.3+1.3i 2.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0-2.0i -1.3+1.3i 2.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0+1.0i -1.3+3.3i -1.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 2.0-3.0i -0.3+4.3i 1.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0+1.0i -3.3+1.3i )  
and
B = ( 2.4-05.0i 2.7+06.9i 3.4+18.2i -6.9-05.3i -14.7+09.7i -6.0-00.6i 31.9-07.7i -3.9+09.3i -1.0+01.6i -3.0+12.2i ) .  
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 (f04ccce.c)

10.2 Program Data

Program Data (f04ccce.d)

10.3 Program Results

Program Results (f04ccce.r)