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NAG Toolbox: nag_lapack_zgbsv (f07bn)

Purpose

nag_lapack_zgbsv (f07bn) computes the solution to a complex system of linear equations
AX = B ,
AX=B ,
where AA is an nn by nn band matrix, with klkl subdiagonals and kuku superdiagonals, and XX and BB are nn by rr 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 LULU decomposition with partial pivoting and row interchanges to factor AA as A = PLUA=PLU, where PP is a permutation matrix, LL is a product of permutation and unit lower triangular matrices with klkl subdiagonals, and UU is upper triangular with (kl + ku)(kl+ku) superdiagonals. The factored form of AA is then used to solve the system of equations AX = BAX=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
klkl, the number of subdiagonals within the band of the matrix AA.
Constraint: kl0kl0.
2:     ku – int64int32nag_int scalar
kuku, the number of superdiagonals within the band of the matrix AA.
Constraint: ku0ku0.
3:     ab(ldab, : :) – complex array
The first dimension of the array ab must be at least 2 × kl + ku + 12×kl+ku+1
The second dimension of the array must be at least max (1,n)max(1,n)
The nn by nn coefficient matrix AA.
The matrix is stored in rows kl + 1kl+1 to 2kl + ku + 12kl+ku+1; the first klkl rows need not be set, more precisely, the element AijAij must be stored in
ab(kl + ku + 1 + ij,j) = Aij  for ​max (1,jku)imin (n,j + kl).
abkl+ku+1+i-jj=Aij  for ​max(1,j-ku)imin(n,j+kl).
See Section [Further Comments] for further details.
4:     b(ldb, : :) – complex array
The first dimension of the array b must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,nrhs)max(1,nrhs)
The nn by rr right-hand side matrix BB.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b.
nn, the number of linear equations, i.e., the order of the matrix AA.
Constraint: n0n0.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
rr, the number of right-hand sides, i.e., the number of columns of the matrix BB.
Constraint: nrhs0nrhs0.

Input Parameters Omitted from the MATLAB Interface

ldab ldb

Output Parameters

1:     ab(ldab, : :) – complex array
The first dimension of the array ab will be 2 × kl + ku + 12×kl+ku+1
The second dimension of the array will be max (1,n)max(1,n)
ldab2 × kl + ku + 1ldab2×kl+ku+1.
If info0info0, ab stores details of the factorization.
The upper triangular band matrix UU, with kl + kukl+ku superdiagonals, is stored in rows 11 to kl + ku + 1kl+ku+1 of the array, and the multipliers used to form the matrix LL are stored in rows kl + ku + 2kl+ku+2 to 2kl + ku + 12kl+ku+1.
2:     ipiv(n) – int64int32nag_int array
If no constraints are violated, the pivot indices that define the permutation matrix PP; at the iith step row ii of the matrix was interchanged with row ipiv(i)ipivi. ipiv(i) = iipivi=i indicates a row interchange was not required.
3:     b(ldb, : :) – complex array
The first dimension of the array b will be max (1,n)max(1,n)
The second dimension of the array will be max (1,nrhs)max(1,nrhs)
ldbmax (1,n)ldbmax(1,n).
If INFO = 0INFO=0, the nn by rr solution matrix XX.
4:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [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 = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: n, 2: kl, 3: ku, 4: nrhs_p, 5: ab, 6: ldab, 7: ipiv, 8: b, 9: ldb, 10: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
W INFO > 0INFO>0
If info = iinfo=i, uiiuii is exactly zero. The factorization has been completed, but the factor UU 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) = b ,
(A+E) x^ = b ,
where
E1 = O(ε) A1
E1 = O(ε) A1
and ε ε  is the machine precision. An approximate error bound for the computed solution is given by
(x1)/(x1) κ(A) (E1)/(A1) ,
x^-x 1 x1 κ(A) E1 A1 ,
where κ(A) = A11 A1 κ(A) = A-11 A1 , the condition number of A 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 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 n=6 , kl = 1 kl=1 , and ku = 2 ku=2 . Storage of the band matrix A 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 *
* * * + + + * * 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 u14 , u25 u25  and u36 u36 .
The total number of floating point operations required to solve the equations AX = B AX=B  depends upon the pivoting required, but if nkl + ku nkl + ku  then it is approximately bounded by O( nkl ( kl + ku ) ) O( nkl ( kl + ku ) )  for the factorization and O( n ( 2 kl + ku ) r ) O( n ( 2 kl + ku ) r )  for the solution following the factorization.
The real analogue of this function is nag_lapack_dgbsv (f07ba).

Example

function nag_lapack_zgbsv_example
kl = int64(1);
ku = int64(2);
ab = [complex(0),  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];
[abOut, ipiv, bOut, info] = nag_lapack_zgbsv(kl, ku, ab, b)
 

abOut =

   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.5900 - 0.4800i
   0.0000 + 0.0000i   0.0000 + 0.0000i  -3.9900 + 4.0100i   3.3300 - 1.0400i
   0.0000 + 0.0000i  -1.4800 - 1.7500i  -1.0600 + 1.9400i  -1.7692 - 1.8587i
   0.0000 + 6.3000i  -0.7700 + 2.8300i   4.9303 - 3.0086i   0.4338 + 0.1233i
   0.3587 + 0.2619i   0.2314 + 0.6358i   0.7604 + 0.2429i   0.0000 + 0.0000i


ipiv =

                    2
                    3
                    3
                    4


bOut =

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


info =

                    0


function f07bn_example
kl = int64(1);
ku = int64(2);
ab = [complex(0),  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];
[abOut, ipiv, bOut, info] = f07bn(kl, ku, ab, b)
 

abOut =

   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.5900 - 0.4800i
   0.0000 + 0.0000i   0.0000 + 0.0000i  -3.9900 + 4.0100i   3.3300 - 1.0400i
   0.0000 + 0.0000i  -1.4800 - 1.7500i  -1.0600 + 1.9400i  -1.7692 - 1.8587i
   0.0000 + 6.3000i  -0.7700 + 2.8300i   4.9303 - 3.0086i   0.4338 + 0.1233i
   0.3587 + 0.2619i   0.2314 + 0.6358i   0.7604 + 0.2429i   0.0000 + 0.0000i


ipiv =

                    2
                    3
                    3
                    4


bOut =

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


info =

                    0



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

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