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NAG Toolbox: nag_lapack_zgtsv (f07cn)

Purpose

nag_lapack_zgtsv (f07cn) computes the solution to a complex system of linear equations
AX = B ,
AX=B ,
where AA is an nn by nn tridiagonal matrix and XX and BB are nn by rr matrices.

Syntax

[dl, d, du, b, info] = f07cn(dl, d, du, b, 'n', n, 'nrhs_p', nrhs_p)
[dl, d, du, b, info] = nag_lapack_zgtsv(dl, d, du, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zgtsv (f07cn) uses Gaussian elimination with partial pivoting and row interchanges to solve the equations AX = B AX=B . The matrix A A  is factorized as A = PLU A=PLU , where P P  is a permutation matrix, L L  is unit lower triangular with at most one nonzero subdiagonal element per column, and U U  is an upper triangular band matrix, with two superdiagonals.
Note that the equations ATX = BATX=B may be solved by interchanging the order of the arguments du and dl.

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

Parameters

Compulsory Input Parameters

1:     dl( : :) – complex array
Note: the dimension of the array dl must be at least max (1,n1)max(1,n-1).
Must contain the (n1)(n-1) subdiagonal elements of the matrix AA.
2:     d( : :) – complex array
Note: the dimension of the array d must be at least max (1,n)max(1,n).
Must contain the nn diagonal elements of the matrix AA.
3:     du( : :) – complex array
Note: the dimension of the array du must be at least max (1,n1)max(1,n-1).
Must contain the (n1)(n-1) superdiagonal elements of the matrix AA.
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)
Note: to solve the equations Ax = bAx=b, where bb is a single right-hand side, b may be supplied as a one-dimensional array with length ldb = max (1,n)ldb=max(1,n).
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 The dimension of the array d.
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

ldb

Output Parameters

1:     dl( : :) – complex array
Note: the dimension of the array dl must be at least max (1,n1)max(1,n-1).
If no constrains are violated, dl stores the (n2n-2) elements of the second superdiagonal of the upper triangular matrix UU from the LULU factorization of AA, in dl(1),dl(2),,dl(n2)dl1,dl2,,dln-2.
2:     d( : :) – complex array
Note: the dimension of the array d must be at least max (1,n)max(1,n).
If no constraints are violated, d stores the nn diagonal elements of the upper triangular matrix UU from the LULU factorization of AA.
3:     du( : :) – complex array
Note: the dimension of the array du must be at least max (1,n1)max(1,n-1).
If no constraints are violated, du stores the (n1)(n-1) elements of the first superdiagonal of UU.
4:     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)
Note: to solve the equations Ax = bAx=b, where bb is a single right-hand side, b may be supplied as a one-dimensional array with length ldb = max (1,n)ldb=max(1,n).
ldbmax (1,n)ldbmax(1,n).
If INFO = 0INFO=0, the nn by rr solution matrix XX.
5:     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: nrhs_p, 3: dl, 4: d, 5: du, 6: b, 7: ldb, 8: 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, and the solution has not been computed. The factorization has not been completed unless i = ni=n.

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.
Alternatives to nag_lapack_zgtsv (f07cn), which return condition and error estimates are nag_linsys_complex_tridiag_solve (f04cc) and nag_lapack_zgtsvx (f07cp).

Further Comments

The total number of floating point operations required to solve the equations AX = B AX=B  is proportional to nr nr .
The real analogue of this function is nag_lapack_dgtsv (f07ca).

Example

function nag_lapack_zgtsv_example
dl = [ 1 - 2i;
      1 + 1i;
      2 - 3i;
      1 + 1i];
d = [ -1.3 + 1.3i;
      -1.3 + 1.3i;
      -1.3 + 3.3i;
      -0.3 + 4.3i;
      -3.3 + 1.3i];
du = [ 2 - 1i;
      2 + 1i;
      -1 + 1i;
      1 - 1i];
b = [ 2.4 - 5i;
      3.4 + 18.2i;
      -14.7 + 9.7i;
      31.9 - 7.7i;
      -1 + 1.6i];
[dlOut, dOut, duOut, bOut, info] = nag_lapack_zgtsv(dl, d, du, b)
 

dlOut =

   2.0000 + 1.0000i
  -1.0000 + 1.0000i
   1.0000 - 1.0000i
   1.0000 + 1.0000i


dOut =

   1.0000 - 2.0000i
   1.0000 + 1.0000i
   2.0000 - 3.0000i
   1.0000 + 1.0000i
  -1.3399 + 0.2875i


duOut =

  -1.3000 + 1.3000i
  -1.3000 + 3.3000i
  -0.3000 + 4.3000i
  -3.3000 + 1.3000i


bOut =

   1.0000 + 1.0000i
   3.0000 - 1.0000i
   4.0000 + 5.0000i
  -1.0000 - 2.0000i
   1.0000 - 1.0000i


info =

                    0


function f07cn_example
dl = [ 1 - 2i;
      1 + 1i;
      2 - 3i;
      1 + 1i];
d = [ -1.3 + 1.3i;
      -1.3 + 1.3i;
      -1.3 + 3.3i;
      -0.3 + 4.3i;
      -3.3 + 1.3i];
du = [ 2 - 1i;
      2 + 1i;
      -1 + 1i;
      1 - 1i];
b = [ 2.4 - 5i;
      3.4 + 18.2i;
      -14.7 + 9.7i;
      31.9 - 7.7i;
      -1 + 1.6i];
[dlOut, dOut, duOut, bOut, info] = f07cn(dl, d, du, b)
 

dlOut =

   2.0000 + 1.0000i
  -1.0000 + 1.0000i
   1.0000 - 1.0000i
   1.0000 + 1.0000i


dOut =

   1.0000 - 2.0000i
   1.0000 + 1.0000i
   2.0000 - 3.0000i
   1.0000 + 1.0000i
  -1.3399 + 0.2875i


duOut =

  -1.3000 + 1.3000i
  -1.3000 + 3.3000i
  -0.3000 + 4.3000i
  -3.3000 + 1.3000i


bOut =

   1.0000 + 1.0000i
   3.0000 - 1.0000i
   4.0000 + 5.0000i
  -1.0000 - 2.0000i
   1.0000 - 1.0000i


info =

                    0



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

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