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NAG Toolbox: nag_linsys_complex_band_solve (f04cb)

Purpose

nag_linsys_complex_band_solve (f04cb) computes the solution to a complex system of linear equations AX = BAX=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. An estimate of the condition number of AA and an error bound for the computed solution are also returned.

Syntax

[ab, ipiv, b, rcond, errbnd, ifail] = f04cb(kl, ku, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[ab, ipiv, b, rcond, errbnd, ifail] = nag_linsys_complex_band_solve(kl, ku, ab, b, 'n', n, 'nrhs_p', nrhs_p)

Description

The LULU decomposition with partial pivoting and row interchanges is used to factor AA as A = PLUA=PLU, where PP is a permutation matrix, LL is the product of permutation matrices 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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

Parameters

Compulsory Input Parameters

1:     kl – int64int32nag_int scalar
The number of subdiagonals klkl, within the band of AA.
Constraint: kl0kl0.
2:     ku – int64int32nag_int scalar
The number of superdiagonals kuku, within the band of 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 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 matrix of right-hand sides BB.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b The second dimension of the array ab.
The number of linear equations nn, i.e., the order of the matrix AA.
Constraint: n0n0.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
The number of right-hand sides rr, 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 ifail0ifail0, 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 ifail0ifail0, 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 ifail = 0ifail=0 or n + 1n+1, the nn by rr solution matrix XX.
4:     rcond – double scalar
If ifail0ifail0, an estimate of the reciprocal of the condition number of the matrix AA, computed as rcond = (A1A11)rcond=(A1A-11).
5:     errbnd – double scalar
If ifail = 0ifail=0 or n + 1n+1, an estimate of the forward error bound for a computed solution x^, such that x1 / x1errbndx^-x1/x1errbnd, where x^ is a column of the computed solution returned in the array b and xx is the corresponding column of the exact solution XX. If rcond is less than machine precision, then errbnd is returned as unity.
6:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:

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

  ifail < 0andifail999ifail<0andifail-999
If ifail = iifail=-i, the iith argument had an illegal value.
  ifail = 999ifail=-999
Allocation of memory failed. The double allocatable memory required is n, and the complex allocatable memory required is 2 × n2×n. In this case the factorization and the solution XX have been computed, but rcond and errbnd have not been computed.
  ifail > 0andifailNifail>0andifailN
If ifail = iifail=i, uiiuii is exactly zero. The factorization has been completed, but the factor UU is exactly singular, so the solution could not be computed.
W ifail = N + 1ifail=N+1
rcond is less than machine precision, so that the matrix AA is numerically singular. A solution to the equations AX = BAX=B has nevertheless been 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 εε is the machine precision. An approximate error bound for the computed solution is given by
(x1)/(x1) κ(A) (E1)/(A1) ,
x^-x1 x1 κ(A) E1 A1 ,
where κ(A) = A11A1κ(A)=A-11A1, the condition number of AA with respect to the solution of the linear equations. nag_linsys_complex_band_solve (f04cb) uses the approximation E1 = εA1E1=εA1 to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

The band storage scheme for the array ab is illustrated by the following example, when n = 6n=6, kl = 1kl=1, and ku = 2ku=2. Storage of the band matrix AA 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 u14u14, u25u25 and u36u36.
The total number of floating point operations required to solve the equations AX = BAX=B depends upon the pivoting required, but if nkl + kunkl+ku then it is approximately bounded by O(nkl(kl + ku)) O( n kl ( kl + ku ) )  for the factorization and O( n (2kl + ku) ,r) O( n ( 2 kl + ku ) ,r)  for the solution following the factorization. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The real analogue of nag_linsys_complex_band_solve (f04cb) is nag_linsys_real_band_solve (f04bb).

Example

function nag_linsys_complex_band_solve_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,  12.85 + 2.84i;
      -22.72 - 53.9i,  -70.22 + 21.57i;
      28.24 - 38.6i,  -20.73 - 1.23i;
      -34.56 + 16.73i,  26.01 + 31.97i];
[abOut, ipiv, bOut, rcond, errbnd, ifail] = nag_linsys_complex_band_solve(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 + 6.0000i
   1.0000 - 7.0000i  -7.0000 - 4.0000i
  -5.0000 + 4.0000i   3.0000 + 5.0000i
   6.0000 - 8.0000i  -8.0000 + 2.0000i


rcond =

    0.0096


errbnd =

   1.1572e-14


ifail =

                    0


function f04cb_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,  12.85 + 2.84i;
      -22.72 - 53.9i,  -70.22 + 21.57i;
      28.24 - 38.6i,  -20.73 - 1.23i;
      -34.56 + 16.73i,  26.01 + 31.97i];
[abOut, ipiv, bOut, rcond, errbnd, ifail] = f04cb(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 + 6.0000i
   1.0000 - 7.0000i  -7.0000 - 4.0000i
  -5.0000 + 4.0000i   3.0000 + 5.0000i
   6.0000 - 8.0000i  -8.0000 + 2.0000i


rcond =

    0.0096


errbnd =

   1.1572e-14


ifail =

                    0



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