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NAG Toolbox: nag_lapack_zpttrs (f07js)

Purpose

nag_lapack_zpttrs (f07js) computes the solution to a complex system of linear equations AX = B AX=B , where A A  is an n n  by n n  Hermitian positive definite tridiagonal matrix and X X  and B B  are n n  by r r  matrices, using the LDLH LDLH  factorization returned by nag_lapack_zpttrf (f07jr).

Syntax

[b, info] = f07js(uplo, d, e, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zpttrs(uplo, d, e, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zpttrs (f07js) should be preceded by a call to nag_lapack_zpttrf (f07jr), which computes a modified Cholesky factorization of the matrix A A  as
A = LDLH ,
A=LDLH ,
where L L  is a unit lower bidiagonal matrix and D D  is a diagonal matrix, with positive diagonal elements. nag_lapack_zpttrs (f07js) then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form UHDU UHDU , where U U  is a unit upper bidiagonal matrix.

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:     uplo – string (length ≥ 1)
Specifies the form of the factorization as follows:
uplo = 'U'uplo='U'
A = UHDUA=UHDU.
uplo = 'L'uplo='L'
A = LDLHA=LDLH.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     d( : :) – double 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 diagonal matrix DD from the LDLHLDLH or UHDUUHDU factorization of AA.
3:     e( : :) – complex array
Note: the dimension of the array e must be at least max (1,n1)max(1,n-1).
If uplo = 'U'uplo='U', e must contain the (n1)(n-1) superdiagonal elements of the unit upper bidiagonal matrix UU from the UHDUUHDU factorization of AA.
If uplo = 'L'uplo='L', e must contain the (n1)(n-1) subdiagonal elements of the unit lower bidiagonal matrix LL from the LDLHLDLH factorization of 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)
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 dimension of the array d.
nn, 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:     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).
The nn by rr solution matrix XX.
2:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

  info = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: nrhs_p, 4: d, 5: e, 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.

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 this function nag_lapack_zptcon (f07ju) can be used to estimate the condition number of A A  and nag_lapack_zptrfs (f07jv) can be used to obtain approximate error bounds.

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_dpttrs (f07je).

Example

function nag_lapack_zpttrs_example
uplo = 'U';
d = [16;
     9;
     1;
     4];
e = [ 1 - 1i;
      2 + 1i;
      1 + 4i];
b = [ 64 + 16i,  -16 - 32i;
      93 + 62i,  61 - 66i;
      78 - 80i,  71 - 74i;
      14 - 27i,  35 + 15i];
[bOut, info] = nag_lapack_zpttrs(uplo, d, e, b)
 

bOut =

   2.0000 + 1.0000i  -3.0000 - 2.0000i
   1.0000 + 1.0000i   1.0000 + 1.0000i
   1.0000 - 2.0000i   1.0000 - 2.0000i
   1.0000 - 1.0000i   2.0000 + 1.0000i


info =

                    0


function f07js_example
uplo = 'U';
d = [16;
     9;
     1;
     4];
e = [ 1 - 1i;
      2 + 1i;
      1 + 4i];
b = [ 64 + 16i,  -16 - 32i;
      93 + 62i,  61 - 66i;
      78 - 80i,  71 - 74i;
      14 - 27i,  35 + 15i];
[bOut, info] = f07js(uplo, d, e, b)
 

bOut =

   2.0000 + 1.0000i  -3.0000 - 2.0000i
   1.0000 + 1.0000i   1.0000 + 1.0000i
   1.0000 - 2.0000i   1.0000 - 2.0000i
   1.0000 - 1.0000i   2.0000 + 1.0000i


info =

                    0



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Chapter Introduction
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