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NAG Toolbox: nag_lapack_dgbsv (f07ba)

Purpose

nag_lapack_dgbsv (f07ba) computes the solution to a real 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] = f07ba(kl, ku, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[ab, ipiv, b, info] = nag_lapack_dgbsv(kl, ku, ab, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dgbsv (f07ba) 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, : :) – double 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, : :) – double 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, : :) – double 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, : :) – double 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_dgbsv (f07ba), nag_lapack_dgbcon (f07bg) can be used to estimate the condition number of A A  and nag_lapack_dgbrfs (f07bh) can be used to obtain approximate error bounds. Alternatives to nag_lapack_dgbsv (f07ba), which return condition and error estimates directly are nag_linsys_real_band_solve (f04bb) and nag_lapack_dgbsvx (f07bb).

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 complex analogue of this function is nag_lapack_zgbsv (f07bn).

Example

function nag_lapack_dgbsv_example
kl = int64(1);
ku = int64(2);
ab = [0, 0, 0, 0;
     0, 0, -3.66, -2.13;
     0, 2.54, -2.73, 4.07;
     -0.23, 2.46, 2.46, -3.82;
     -6.98, 2.56, -4.78, 0];
b = [4.42;
     27.13;
     -6.14;
     10.5];
[abOut, ipiv, bOut, info] = nag_lapack_dgbsv(kl, ku, ab, b)
 

abOut =

         0         0         0   -2.1300
         0         0   -2.7300    4.0700
         0    2.4600    2.4600   -3.8391
   -6.9800    2.5600   -5.9329   -0.7269
    0.0330    0.9605    0.8057         0


ipiv =

                    2
                    3
                    3
                    4


bOut =

   -2.0000
    3.0000
    1.0000
   -4.0000


info =

                    0


function f07ba_example
kl = int64(1);
ku = int64(2);
ab = [0, 0, 0, 0;
     0, 0, -3.66, -2.13;
     0, 2.54, -2.73, 4.07;
     -0.23, 2.46, 2.46, -3.82;
     -6.98, 2.56, -4.78, 0];
b = [4.42;
     27.13;
     -6.14;
     10.5];
[abOut, ipiv, bOut, info] = f07ba(kl, ku, ab, b)
 

abOut =

         0         0         0   -2.1300
         0         0   -2.7300    4.0700
         0    2.4600    2.4600   -3.8391
   -6.9800    2.5600   -5.9329   -0.7269
    0.0330    0.9605    0.8057         0


ipiv =

                    2
                    3
                    3
                    4


bOut =

   -2.0000
    3.0000
    1.0000
   -4.0000


info =

                    0



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