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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgbtrf (f07br)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_zgbtrf (f07br) computes the LU factorization of a complex m by n band matrix.


[ab, ipiv, info] = f07br(m, kl, ku, ab, 'n', n)
[ab, ipiv, info] = nag_lapack_zgbtrf(m, kl, ku, ab, 'n', n)


nag_lapack_zgbtrf (f07br) forms the LU factorization of a complex m by n band matrix A using partial pivoting, with row interchanges. Usually m=n, and then, if A has kl nonzero subdiagonals and ku nonzero superdiagonals, the factorization has the form A=PLU, where P is a permutation matrix, L is a lower triangular matrix with unit diagonal elements and at most kl nonzero elements in each column, and U is an upper triangular band matrix with kl+ku superdiagonals.
Note that L is not a band matrix, but the nonzero elements of L can be stored in the same space as the subdiagonal elements of A. U is a band matrix but with kl additional superdiagonals compared with A. These additional superdiagonals are created by the row interchanges.


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


Compulsory Input Parameters

1:     m int64int32nag_int scalar
m, the number of rows of the matrix A.
Constraint: m0.
2:     kl int64int32nag_int scalar
kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
3:     ku int64int32nag_int scalar
ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
4:     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 m by n 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-kuiminm,j+kl. 
See Further Comments in nag_lapack_zgbsv (f07bn) for further details.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the second dimension of the array ab.
n, the number of columns of the matrix A.
Constraint: n0.

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:     ipivminm,n int64int32nag_int array
The pivot indices that define the permutation matrix. At the ith step, if ipivi>i then row i of the matrix A was interchanged with row ipivi, for i=1,2,,minm,n. ipivii indicates that, at the ith step, a row interchange was not required.
3:     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.

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, and division by zero will occur if it is used to solve a system of equations.


The computed factors L and U are the exact factors of a perturbed matrix A+E, where
EckεPLU ,  
ck is a modest linear function of k=kl+ku+1, and ε is the machine precision. This assumes k minm,n .

Further Comments

The total number of real floating-point operations varies between approximately 8nklku+1 and 8nklkl+ku+1, depending on the interchanges, assuming m=nkl and nku.
A call to nag_lapack_zgbtrf (f07br) may be followed by calls to the functions:
The real analogue of this function is nag_lapack_dgbtrf (f07bd).


This example computes the LU factorization of the matrix A, where
A= -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00+0.00i 0.00+6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00+0.00i -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00+0.00i 0.00+0.00i 4.48-1.09i -0.46-1.72i .  
Here A is treated as a band matrix with one subdiagonal and two superdiagonals.
function f07br_example

fprintf('f07br 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];

[abf, ipiv, info] = f07br( ...
                          m, kl, ku, ab);

mtitle = 'Details of factorization';
[ifail] = x04de( ...
                 m, m, kl, kl+ku, abf, mtitle);

disp('Pivot indices');

f07br example results

 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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