F07JNF (ZPTSV) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F07JNF (ZPTSV) computes the solution to a complex system of linear equations
AX=B ,
where A is an n by n Hermitian positive definite tridiagonal matrix, and X and B are n by r matrices.

2  Specification

REAL (KIND=nag_wp)  D(*)
COMPLEX (KIND=nag_wp)  E(*), B(LDB,*)
The routine may be called by its LAPACK name zptsv.

3  Description

F07JNF (ZPTSV) factors A as A=LDLH. 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 http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Parameters

1:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
2:     NRHS – INTEGERInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: NRHS0.
3:     D(*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array D must be at least max1,N.
On entry: the n diagonal elements of the tridiagonal matrix A.
On exit: the n diagonal elements of the diagonal matrix D from the factorization A=LDLH.
4:     E(*) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array E must be at least max1,N-1.
On entry: the n-1 subdiagonal elements of the tridiagonal matrix A.
On exit: the n-1 subdiagonal elements of the unit bidiagonal factor L from the LDLH factorization of A. (E can also be regarded as the superdiagonal of the unit bidiagonal factor U from the UHDU factorization of A.)
5:     B(LDB,*) – 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 right-hand side matrix B.
On exit: if INFO=0, the n by r solution matrix X.
6:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07JNF (ZPTSV) is called.
Constraint: LDBmax1,N.
7:     INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
If INFO=-i, the ith argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
If INFO=i, the leading minor of order i is not positive definite, and the solution has not been computed. The factorization has not been completed unless i=N.

7  Accuracy

The computed solution for a single right-hand side, x^ , satisfies an equation of the form
A+E x^=b ,
E1 = Oε A1
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κA 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.
F07JPF (ZPTSVX) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, F04CGF solves Ax=b  and returns a forward error bound and condition estimate. F04CGF calls F07JNF (ZPTSV) to solve the equations.

8  Further Comments

The number of floating point operations required for the factorization of A  is proportional to n , and the number of floating point operations required for the solution of the equations is proportional to nr , where r  is the number of right-hand sides.
The real analogue of this routine is F07JAF (DPTSV).

9  Example

This example solves the equations
Ax=b ,
where A  is the Hermitian positive definite tridiagonal matrix
A = 16.0i+00.0 16.0-16.0i 0.0i+0.0 0.0i+0.0 16.0+16.0i 41.0i+00.0 18.0+9.0i 0.0i+0.0 0.0i+00.0 18.0-09.0i 46.0i+0.0 1.0+4.0i 0.0i+00.0 0.0i+00.0 1.0-4.0i 21.0i+0.0
b = 64.0+16.0i 93.0+62.0i 78.0-80.0i 14.0-27.0i .
Details of the LDLH  factorization of A  are also output.

9.1  Program Text

Program Text (f07jnfe.f90)

9.2  Program Data

Program Data (f07jnfe.d)

9.3  Program Results

Program Results (f07jnfe.r)

F07JNF (ZPTSV) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012