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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_zgbtrf (f07br)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_lapack_zgbtrf (f07br) computes the

L U

factorization of a complex

m

n

band matrix.

Syntax

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

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

Description

nag_lapack_zgbtrf (f07br) forms the

L U

factorization of a complex

m

n

band matrix

A

using partial pivoting, with row interchanges. Usually

m = n

, and then, if

A

has

k_{l}

nonzero subdiagonals and

k_{u}

nonzero superdiagonals, the factorization has the form

A = P L U

, where

P

is a permutation matrix,

L

is a lower triangular matrix with unit diagonal elements and at most

k_{l}

nonzero elements in each column, and

U

is an upper triangular band matrix with

k_{l} + k_{u}

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

k_{l}

additional superdiagonals compared with

A

. These additional superdiagonals are created by the row interchanges.

References

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

Parameters

Compulsory Input Parameters

1: $m$ – int64int32nag_int scalar

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

2: $kl$ – int64int32nag_int scalar

k_{l}

, the number of subdiagonals within the band of the matrix

A

Constraint:

kl \geq 0

3: $ku$ – int64int32nag_int scalar

k_{u}

, the number of superdiagonals within the band of the matrix

A

Constraint:

ku \geq 0

4: $ab (ldab :)$ – complex array

The first dimension of the array ab must be at least

2 \times kl + ku + 1

The second dimension of the array ab must be at least

\max (1, n)

The

m

n

matrix

A

The matrix is stored in rows

k_{l} + 1

2 k_{l} + k_{u} + 1

; the first

k_{l}

rows need not be set, more precisely, the element

A_{i j}

must be stored in

ab (k_{l} + k_{u} + 1 + i - j, j) = A_{i j} for ​ \max (1, j - k_{u}) \leq i \leq \min (m, j + k_{l}) .

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: $n \geq 0$ .

Output Parameters

1: $ab (ldab :)$ – complex array: The first dimension of the array ab will be $2 \times kl + ku + 1$ .
The second dimension of the array ab will be $\max (1, n)$ .

If $info \geq 0$ , ab stores details of the factorization.
The upper triangular band matrix $U$ , with $k_{l} + k_{u}$ superdiagonals, is stored in rows $1$ to $k_{l} + k_{u} + 1$ of the array, and the multipliers used to form the matrix $L$ are stored in rows $k_{l} + k_{u} + 2$ to $2 k_{l} + k_{u} + 1$ .
2: $ipiv (\min (m, n))$ – int64int32nag_int array: The pivot indices that define the permutation matrix. At the $i$ th step, if $ipiv (i) > i$ then row $i$ of the matrix $A$ was interchanged with row $ipiv (i)$ , for $i = 1, 2, \dots, \min (m, n)$ . $ipiv (i) \leq i$ indicates that, at the $i$ th 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.

$info < 0$: 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, and division by zero will occur if it is used to solve a system of equations.

Accuracy

The computed factors

L

and

U

are the exact factors of a perturbed matrix

A + E

, where

|E| \leq c (k) ε P |L| |U|,

c (k)

is a modest linear function of

k = k_{l} + k_{u} + 1

, and

ε

is the machine precision. This assumes

k ≪ \min (m, n)

Further Comments

The total number of real floating-point operations varies between approximately

8 n k_{l} (k_{u} + 1)

and

8 n k_{l} (k_{l} + k_{u} + 1)

, depending on the interchanges, assuming

m = n ≫ k_{l}

and

n ≫ k_{u}

A call to nag_lapack_zgbtrf (f07br) may be followed by calls to the functions:

nag_lapack_zgbtrs (f07bs) to solve $A X = B$ , $A^{T} X = B$ or $A^{H} X = B$ ;
nag_lapack_zgbcon (f07bu) to estimate the condition number of $A$ .

The real analogue of this function is nag_lapack_dgbtrf (f07bd).

Example

This example computes the

L U

factorization of the matrix

A

, where

A = (\begin{array}{r} - 1.65 + 2.26 i & - 2.05 - 0.85 i & 0.97 - 2.84 i & 0.00 + 0.00 i \\ 0.00 + 6.30 i & - 1.48 - 1.75 i & - 3.99 + 4.01 i & 0.59 - 0.48 i \\ 0.00 + 0.00 i & - 0.77 + 2.83 i & - 1.06 + 1.94 i & 3.33 - 1.04 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & 4.48 - 1.09 i & - 0.46 - 1.72 i \end{array}) .

Here

A

is treated as a band matrix with one subdiagonal and two superdiagonals.

Open in the MATLAB editor: f07br_example

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');
disp(double(ipiv'));

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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Chapter Contents

Chapter Introduction

NAG Toolbox