NAG CL Interface
f07cpc (zgtsvx)

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1 Purpose

f07cpc uses the LU factorization to compute the solution to a complex system of linear equations
AX=B ,  ATX=B   or   AHX=B ,  
where A is a tridiagonal matrix of order n 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  f07cpc (Nag_OrderType order, Nag_FactoredFormType fact, Nag_TransType trans, Integer n, Integer nrhs, const Complex dl[], const Complex d[], const Complex du[], Complex dlf[], Complex df[], Complex duf[], Complex du2[], Integer ipiv[], const Complex b[], Integer pdb, Complex x[], Integer pdx, double *rcond, double ferr[], double berr[], NagError *fail)
The function may be called by the names: f07cpc, nag_lapacklin_zgtsvx or nag_zgtsvx.

3 Description

f07cpc performs the following steps:
  1. 1.If fact=Nag_NotFactored, the LU decomposition is used to factor the matrix A as A=LU, where L is a product of permutation and unit lower bidiagonal matrices and U is upper triangular with nonzeros in only the main diagonal and first two superdiagonals.
  2. 2.If some uii=0, so that U is exactly singular, then the function returns with fail.errnum=i. 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.
  3. 3.The system of equations is solved for X using the factored form of A.
  4. 4.Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

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 has been supplied.
fact=Nag_Factored
dlf, df, duf, du2 and ipiv contain the factorized form of the matrix A. dlf, df, duf, du2 and ipiv will not be modified.
fact=Nag_NotFactored
The matrix A will be copied to dlf, df and duf and factorized.
Constraint: fact=Nag_Factored or Nag_NotFactored.
3: trans Nag_TransType Input
On entry: specifies the form of the system of equations.
trans=Nag_NoTrans
AX=B (No transpose).
trans=Nag_Trans
ATX=B (Transpose).
trans=Nag_ConjTrans
AHX=B (Conjugate transpose).
Constraint: trans=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
4: n Integer Input
On entry: n, 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: dl[dim] const Complex Input
Note: the dimension, dim, of the array dl must be at least max(1,n-1).
On entry: the (n-1) subdiagonal elements of A.
7: d[dim] const Complex Input
Note: the dimension, dim, of the array d must be at least max(1,n).
On entry: the n diagonal elements of A.
8: du[dim] const Complex Input
Note: the dimension, dim, of the array du must be at least max(1,n-1).
On entry: the (n-1) superdiagonal elements of A.
9: dlf[dim] Complex Input/Output
Note: the dimension, dim, of the array dlf must be at least max(1,n-1).
On entry: if fact=Nag_Factored, dlf contains the (n-1) multipliers that define the matrix L from the LU factorization of A.
On exit: if fact=Nag_NotFactored, dlf contains the (n-1) multipliers that define the matrix L from the LU factorization of A.
10: df[dim] Complex Input/Output
Note: the dimension, dim, of the array df must be at least max(1,n).
On entry: if fact=Nag_Factored, df contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
On exit: if fact=Nag_NotFactored, df contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
11: duf[dim] Complex Input/Output
Note: the dimension, dim, of the array duf must be at least max(1,n-1).
On entry: if fact=Nag_Factored, duf contains the (n-1) elements of the first superdiagonal of U.
On exit: if fact=Nag_NotFactored, duf contains the (n-1) elements of the first superdiagonal of U.
12: du2[dim] Complex Input/Output
Note: the dimension, dim, of the array du2 must be at least max(1,n-2).
On entry: if fact=Nag_Factored, du2 contains the (n-2) elements of the second superdiagonal of U.
On exit: if fact=Nag_NotFactored, du2 contains the (n-2) elements of the second superdiagonal of U.
13: ipiv[dim] Integer Input/Output
Note: the dimension, dim, of the array ipiv must be at least max(1,n).
On entry: if fact=Nag_Factored, ipiv contains the pivot indices from the LU factorization of A.
On exit: if fact=Nag_NotFactored, ipiv contains the pivot indices from the LU factorization of A; 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.
14: b[dim] const Complex Input
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.
15: 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).
16: x[dim] Complex 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.
17: 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).
18: rcond double * Output
On exit: the estimate of the reciprocal condition number of the matrix A. If rcond=0.0, the matrix may be exactly singular. This condition is indicated by fail.code= NE_SINGULAR. Otherwise, if rcond is less than the machine precision, the matrix is singular to working precision. This condition is indicated by fail.code= NE_SINGULAR_WP.
19: 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.
20: 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).
21: 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_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
Element value of the diagonal is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. rcond=0.0 is returned.
Element value of the diagonal is exactly zero. The factorization has not been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. rcond=0.0 is returned.
NE_SINGULAR_WP
U 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
|E| c (n) ε |L| |U| ,  
c(n) is a modest linear function of n, and ε is the machine precision. See Section 9.3 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

f07cpc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07cpc 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. The solution is then refined, and the errors estimated, using iterative refinement.
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 this function is f07cbc.

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 ) .  
Estimates for the backward errors, forward errors and condition number are also output.

10.1 Program Text

Program Text (f07cpce.c)

10.2 Program Data

Program Data (f07cpce.d)

10.3 Program Results

Program Results (f07cpce.r)