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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgbsv (f07bn)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zgbsv (f07bn) computes the solution to a complex system of linear equations
AX=B ,  
where A is an n by n band matrix, with kl subdiagonals and ku superdiagonals, and X and B are n by r matrices.

Syntax

[ab, ipiv, b, info] = f07bn(kl, ku, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[ab, ipiv, b, info] = nag_lapack_zgbsv(kl, ku, ab, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zgbsv (f07bn) uses the LU decomposition with partial pivoting and row interchanges to factor A as A=PLU, where P is a permutation matrix, L is a product of permutation and unit lower triangular matrices with kl subdiagonals, and U is upper triangular with kl+ku superdiagonals. The factored form of A is then used to solve the system of equations AX=B.

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

Parameters

Compulsory Input Parameters

1:     kl int64int32nag_int scalar
kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
2:     ku int64int32nag_int scalar
ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
3:     abldab: – complex array
The first dimension of the array ab must be at least 2×kl+ku+1.
The second dimension of the array ab must be at least max1,n.
The n by n coefficient matrix A.
The matrix is stored in rows kl+1 to 2kl+ku+1; the first kl rows need not be set, more precisely, the element Aij must be stored in
abkl+ku+1+i-jj=Aij  for ​max1,j-kuiminn,j+kl. 
See Further Comments for further details.
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 right-hand side matrix B.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array b.
n, the number of linear equations, i.e., 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:     abldab: – complex array
The first dimension of the array ab will be 2×kl+ku+1.
The second dimension of the array ab will be max1,n.
If info0, ab stores details of the factorization.
The upper triangular band matrix U, with kl+ku superdiagonals, is stored in rows 1 to kl+ku+1 of the array, and the multipliers used to form the matrix L are stored in rows kl+ku+2 to 2kl+ku+1.
2:     ipivn int64int32nag_int array
If no constraints are violated, the pivot indices that define the permutation matrix P; at the ith step row i of the matrix was interchanged with row ipivi. ipivi=i indicates a row interchange was not required.
3:     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.
If info=0, the n by r solution matrix X.
4:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  info>0
Element _ of the diagonal is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed.

Accuracy

The computed solution for a single right-hand side, x^ , satisfies an equation of the form
A+E x^ = b ,  
where
E1 = Oε A1  
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 nag_lapack_zgbsv (f07bn), nag_lapack_zgbcon (f07bu) can be used to estimate the condition number of A  and nag_lapack_zgbrfs (f07bv) can be used to obtain approximate error bounds. Alternatives to nag_lapack_zgbsv (f07bn), which return condition and error estimates directly are nag_linsys_complex_band_solve (f04cb) and nag_lapack_zgbsvx (f07bp).

Further Comments

The band storage scheme for the array ab is illustrated by the following example, when n=6 , kl=1 , and ku=2 . Storage of the band matrix A  in the array ab:
* * * + + + * * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 *  
Array elements marked * need not be set and are not referenced by the function. Array elements marked + need not be set, but are defined on exit from the function and contain the elements u14 , u25  and u36 .
The total number of floating-point operations required to solve the equations AX=B  depends upon the pivoting required, but if nkl + ku  then it is approximately bounded by O nkl kl + ku  for the factorization and O n 2 kl + ku r  for the solution following the factorization.
The real analogue of this function is nag_lapack_dgbsv (f07ba).

Example

This example solves the equations
Ax=b ,  
where A  is the band matrix
A = -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00i+0.00 6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00i+0.00 -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00i+0.00 0.00i+0.00 4.48-1.09i -0.46-1.72i  
and
b = -1.06+21.50i -22.72-53.90i 28.24-38.60i -34.56+16.73i .  
Details of the LU  factorization of A  are also output.
function f07bn_example


fprintf('f07bn example results\n\n');

m  = int64(4);
kl = int64(1);
ku = int64(2);
ab = [ 0    + 0i,     0    + 0i,     0    + 0i,     0    + 0i;
       0    + 0i,     0    + 0i,     0.97 - 2.84i,  0.59 - 0.48i;
       0    + 0i,    -2.05 - 0.85i, -3.99 + 4.01i,  3.33 - 1.04i;
      -1.65 + 2.26i, -1.48 - 1.75i, -1.06 + 1.94i, -0.46 - 1.72i;
       0    + 6.3i,  -0.77 + 2.83i,  4.48 - 1.09i,  0    + 0i];
b  = [ -1.06 + 21.5i;
      -22.72 - 53.9i;
       28.24 - 38.6i;
      -34.56 + 16.73i];

% Solve Ax = B
[abf, ipiv, x, info] = f07bn( ...
                              kl, ku, ab, b);

disp('Solution');
disp(x');
mtitle = 'Details of factorization';
[ifail] = x04de( ...
                 m, m, kl, kl+ku, abf, mtitle);
fprintf('\n');
disp('Pivot indices');
disp(double(ipiv'));


f07bn example results

Solution
  -3.0000 - 2.0000i   1.0000 + 7.0000i  -5.0000 - 4.0000i   6.0000 + 8.0000i

 Details of factorization
             1          2          3          4
 1      0.0000    -1.4800    -3.9900     0.5900
        6.3000    -1.7500     4.0100    -0.4800

 2      0.3587    -0.7700    -1.0600     3.3300
        0.2619     2.8300     1.9400    -1.0400

 3                 0.2314     4.9303    -1.7692
                   0.6358    -3.0086    -1.8587

 4                            0.7604     0.4338
                              0.2429     0.1233

Pivot indices
     2     3     3     4


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