NAG CL Interface
f07jsc (zpttrs)

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1 Purpose

f07jsc computes the solution to a complex system of linear equations AX=B , where A is an n × n Hermitian positive definite tridiagonal matrix and X and B are n × r matrices, using the LDLH factorization returned by f07jrc.

2 Specification

#include <nag.h>
void  f07jsc (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer nrhs, const double d[], const Complex e[], Complex b[], Integer pdb, NagError *fail)
The function may be called by the names: f07jsc, nag_lapacklin_zpttrs or nag_zpttrs.

3 Description

f07jsc should be preceded by a call to f07jrc, which computes a modified Cholesky factorization of the matrix A as
A=LDLH ,  
where L is a unit lower bidiagonal matrix and D is a diagonal matrix, with positive diagonal elements. f07jsc then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form UHDU , 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: uplo Nag_UploType Input
On entry: specifies the form of the factorization as follows:
uplo=Nag_Upper
A=UHDU.
uplo=Nag_Lower
A=LDLH.
Constraint: uplo=Nag_Upper or Nag_Lower.
3: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
4: nrhs Integer Input
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
5: 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 LDLH or UHDU factorization of A.
6: e[dim] const Complex Input
Note: the dimension, dim, of the array e must be at least max(1,n-1).
On entry: if uplo=Nag_Upper, e must contain the (n-1) superdiagonal elements of the unit upper bidiagonal matrix U from the UHDU factorization of A.
If uplo=Nag_Lower, e must contain the (n-1) subdiagonal elements of the unit lower bidiagonal matrix L from the LDLH factorization of A.
7: 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: the n×r solution matrix X.
8: 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).
9: 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 f07juc can be used to estimate the condition number of A and f07jvc can be used to obtain approximate error bounds.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07jsc 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 real analogue of this function is f07jec.

10 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 )  
and
B = ( 64.0+16.0i -16.0-32.0i 93.0+62.0i 61.0-66.0i 78.0-80.0i 71.0-74.0i 14.0-27.0i 35.0+15.0i ) .  

10.1 Program Text

Program Text (f07jsce.c)

10.2 Program Data

Program Data (f07jsce.d)

10.3 Program Results

Program Results (f07jsce.r)