NAG Library Routine Document

f07csf  (zgttrs)


    1  Purpose
    7  Accuracy


f07csf (zgttrs) computes the solution to a complex system of linear equations AX=B  or ATX=B  or AHX=B , where A  is an n  by n  tridiagonal matrix and X  and B  are n  by r  matrices, using the LU  factorization returned by f07crf (zgttrf).


Fortran Interface
Subroutine f07csf ( trans, n, nrhs, dl, d, du, du2, ipiv, b, ldb, info)
Integer, Intent (In):: n, nrhs, ipiv(*), ldb
Integer, Intent (Out):: info
Complex (Kind=nag_wp), Intent (In):: dl(*), d(*), du(*), du2(*)
Complex (Kind=nag_wp), Intent (Inout):: b(ldb,*)
Character (1), Intent (In):: trans
C Header Interface
#include nagmk26.h
void  f07csf_ ( const char *trans, const Integer *n, const Integer *nrhs, const Complex dl[], const Complex d[], const Complex du[], const Complex du2[], const Integer ipiv[], Complex b[], const Integer *ldb, Integer *info, const Charlen length_trans)
The routine may be called by its LAPACK name zgttrs.


f07csf (zgttrs) should be preceded by a call to f07crf (zgttrf), which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix A  as
A=PLU ,  
where P  is a permutation matrix, L  is unit lower triangular with at most one nonzero subdiagonal element in each column, and U  is an upper triangular band matrix, with two superdiagonals. f07csf (zgttrs) then utilizes the factorization to solve the required equations.


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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore


1:     trans – Character(1)Input
On entry: specifies the equations to be solved as follows:
Solve AX=B for X.
Solve ATX=B for X.
Solve AHX=B for X.
Constraint: trans='N', 'T' or 'C'.
2:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
3:     nrhs – IntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
4:     dl* – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array dl must be at least max1,n-1.
On entry: must contain the n-1 multipliers that define the matrix L of the LU factorization of A.
5:     d* – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array d must be at least max1,n.
On entry: must contain the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
6:     du* – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array du must be at least max1,n-1.
On entry: must contain the n-1 elements of the first superdiagonal of U.
7:     du2* – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array du2 must be at least max1,n-2.
On entry: must contain the n-2 elements of the second superdiagonal of U.
8:     ipiv* – Integer arrayInput
Note: the dimension of the array ipiv must be at least max1,n.
On entry: must contain the n pivot indices that define the permutation matrix P. At the ith step, row i of the matrix was interchanged with row ipivi, and ipivi must always be either i or i+1, ipivi=i indicating that a row interchange was not performed.
9:     bldb* – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b must be at least max1,nrhs.
On entry: the n by r matrix of right-hand sides B.
On exit: the n by r solution matrix X.
10:   ldb – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07csf (zgttrs) is called.
Constraint: ldbmax1,n.
11:   info – IntegerOutput
On exit: info=0 unless the routine detects an error (see Section 6).

Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.


The computed solution for a single right-hand side, x^ , satisfies an equation of the form
A+E x^=b ,  
E1 =OεA1  
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^-x 1 x 1 κA E1 A1 ,  
where κA = 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 routine f07cuf (zgtcon) can be used to estimate the condition number of A  and f07cvf (zgtrfs) can be used to obtain approximate error bounds.

Parallelism and Performance

f07csf (zgttrs) is not threaded in any implementation.

Further Comments

The total number of floating-point operations required to solve the equations AX=B  or ATX=B  or AHX=B  is proportional to nr .
The real analogue of this routine is f07cef (dgttrs).


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  
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 .  

Program Text

Program Text (f07csfe.f90)

Program Data

Program Data (f07csfe.d)

Program Results

Program Results (f07csfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017