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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zpttrs (f07js)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_zpttrs (f07js) 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, using the LDLH  factorization returned by nag_lapack_zpttrf (f07jr).


[b, info] = f07js(uplo, d, e, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zpttrs(uplo, d, e, b, 'n', n, 'nrhs_p', nrhs_p)


nag_lapack_zpttrs (f07js) should be preceded by a call to nag_lapack_zpttrf (f07jr), 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. nag_lapack_zpttrs (f07js) 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.


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


Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies the form of the factorization as follows:
Constraint: uplo='U' or 'L'.
2:     d: – double array
The dimension of the array d must be at least max1,n
Must contain the n diagonal elements of the diagonal matrix D from the LDLH or UHDU factorization of A.
3:     e: – complex array
The dimension of the array e must be at least max1,n-1
If uplo='U', e must contain the n-1 superdiagonal elements of the unit upper bidiagonal matrix U from the UHDU factorization of A.
If uplo='L', e must contain the n-1 subdiagonal elements of the unit lower bidiagonal matrix L from the LDLH factorization of A.
4:     bldb: – complex array
The first dimension of the array b must be at least max1,n.
The second dimension of the array b must be at least max1,nrhs_p.
The n by r matrix of right-hand sides B.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array b and the dimension of the array d.
n, the order of the matrix A.
Constraint: n0.
2:     nrhs_p int64int32nag_int scalar
Default: the second dimension of the array b.
r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs_p0.

Output Parameters

1:     bldb: – complex array
The first dimension of the array b will be max1,n.
The second dimension of the array b will be max1,nrhs_p.
The n by r solution matrix X.
2:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

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 x1 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 function nag_lapack_zptcon (f07ju) can be used to estimate the condition number of A  and nag_lapack_zptrfs (f07jv) can be used to obtain approximate error bounds.

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 nag_lapack_dpttrs (f07je).


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 -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 .  
function f07js_example

fprintf('f07js example results\n\n');

% Hermitian tridiagonal A stored as two diagonals
d = [ 16            41          46            21];
e = [ 16 + 16i      18 - 9i      1 - 4i         ];

% Factorize
[df, ef, info] = f07jr( ...
                        d, e);

b = [ 64 + 16i,  -16 - 32i;
      93 + 62i,   61 - 66i;
      78 - 80i,   71 - 74i;
      14 - 27i,   35 + 15i];

% Solve
uplo = 'L';
[x, info] = f07js( ...
                   uplo, df, ef, b);


f07js example results

   2.0000 + 1.0000i  -3.0000 - 2.0000i
   1.0000 + 1.0000i   1.0000 + 1.0000i
   1.0000 - 2.0000i   1.0000 - 2.0000i
   1.0000 - 1.0000i   2.0000 + 1.0000i

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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