hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

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

NAG Toolbox: nag_lapack_zsysvx (f07np)

Purpose

nag_lapack_zsysvx (f07np) uses the diagonal pivoting factorization to compute the solution to a complex system of linear equations
AX = B ,
AX=B ,
where AA is an nn by nn symmetric matrix and XX and BB are nn by rr matrices. Error bounds on the solution and a condition estimate are also provided.

Syntax

[af, ipiv, x, rcond, ferr, berr, info] = f07np(fact, uplo, a, af, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[af, ipiv, x, rcond, ferr, berr, info] = nag_lapack_zsysvx(fact, uplo, a, af, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zsysvx (f07np) performs the following steps:
  1. If fact = 'N'fact='N', the diagonal pivoting method is used to factor AA. The form of the factorization is A = UDUTA=UDUT if uplo = 'U'uplo='U' or A = LDLTA=LDLT if uplo = 'L'uplo='L', where UU (or LL) is a product of permutation and unit upper (lower) triangular matrices, and DD is symmetric and block diagonal with 11 by 11 and 22 by 22 diagonal blocks.
  2. If some dii = 0dii=0, so that DD is exactly singular, then the function returns with info = iinfo=i. Otherwise, the factored form of AA is used to estimate the condition number of the matrix AA. If the reciprocal of the condition number is less than machine precision, infon + 1infon+1 is returned as a warning, but the function still goes on to solve for XX and compute error bounds as described below.
  3. The system of equations is solved for XX using the factored form of AA.
  4. Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

Parameters

Compulsory Input Parameters

1:     fact – string (length ≥ 1)
Specifies whether or not the factorized form of the matrix AA has been supplied.
fact = 'F'fact='F'
af and ipiv contain the factorized form of the matrix AA. af and ipiv will not be modified.
fact = 'N'fact='N'
The matrix AA will be copied to af and factorized.
Constraint: fact = 'F'fact='F' or 'N''N'.
2:     uplo – string (length ≥ 1)
If uplo = 'U'uplo='U', the upper triangle of AA is stored.
If uplo = 'L'uplo='L', the lower triangle of AA is stored.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
3:     a(lda, : :) – complex array
The first dimension of the array a must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,n)max(1,n)
The nn by nn symmetric matrix AA.
  • If uplo = 'U'uplo='U', the upper triangular part of aa must be stored and the elements of the array below the diagonal are not referenced.
  • If uplo = 'L'uplo='L', the lower triangular part of aa must be stored and the elements of the array above the diagonal are not referenced.
4:     af(ldaf, : :) – complex array
The first dimension of the array af must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,n)max(1,n)
If fact = 'F'fact='F', af contains the block diagonal matrix DD and the multipliers used to obtain the factor UU or LL from the factorization a = UDUTa=UDUT or a = LDLTa=LDLT as computed by nag_lapack_zsytrf (f07nr).
5:     ipiv( : :) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)max(1,n).
If fact = 'F'fact='F', ipiv contains details of the interchanges and the block structure of DD, as determined by nag_lapack_zsytrf (f07nr).
  • if ipiv(i) = k > 0ipivi=k>0, diidii is a 11 by 11 pivot block and the iith row and column of AA were interchanged with the kkth row and column;
  • if uplo = 'U'uplo='U' and ipiv(i1) = ipiv(i) = l < 0ipivi-1=ipivi=-l<0,
    (di1,i1di,i1)
    di,i1dii
    di-1,i-1d-i,i-1 d-i,i-1dii  is a 22 by 22 pivot block and the (i1)(i-1)th row and column of AA were interchanged with the llth row and column;
  • if uplo = 'L'uplo='L' and ipiv(i) = ipiv(i + 1) = m < 0ipivi=ipivi+1=-m<0,
    (diidi + 1,i)
    di + 1,idi + 1,i + 1
    diidi+1,idi+1,idi+1,i+1 is a 22 by 22 pivot block and the (i + 1)(i+1)th row and column of AA were interchanged with the mmth row and column.
6:     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 arrays a, af, b The second dimension of the arrays a, af, ipiv.
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

lda ldaf ldb ldx work lwork rwork

Output Parameters

1:     af(ldaf, : :) – complex array
The first dimension of the array af will be max (1,n)max(1,n)
The second dimension of the array will be max (1,n)max(1,n)
ldafmax (1,n)ldafmax(1,n).
If fact = 'N'fact='N', af returns the block diagonal matrix DD and the multipliers used to obtain the factor UU or LL from the factorization a = UDUTa=UDUT or a = LDLTa=LDLT.
2:     ipiv( : :) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)max(1,n).
If fact = 'N'fact='N', ipiv contains details of the interchanges and the block structure of DD, as determined by nag_lapack_zsytrf (f07nr), as described above.
3:     x(ldx, : :) – complex array
The first dimension of the array x will be max (1,n)max(1,n)
The second dimension of the array will be max (1,nrhs)max(1,nrhs)
ldxmax (1,n)ldxmax(1,n).
If INFO = 0INFO=0 or n + 1n+1, the nn by rr solution matrix XX.
4:     rcond – double scalar
The estimate of the reciprocal condition number of the matrix AA. If rcond = 0.0rcond=0.0, the matrix may be exactly singular. This condition is indicated by INFO > 0andINFOnINFO>0andINFOn. Otherwise, if rcond is less than the machine precision, the matrix is singular to working precision. This condition is indicated by infon + 1infon+1.
5:     ferr( : :) – double array
Note: the dimension of the array ferr must be at least max (1,nrhs)max(1,nrhs).
If INFO = 0INFO=0 or n + 1n+1, an estimate of the forward error bound for each computed solution vector, such that jxj / xjferr(j)x^j-xj/xjferrj where jx^j is the jjth column of the computed solution returned in the array x and xjxj is the corresponding column of the exact solution XX. The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
6:     berr( : :) – double array
Note: the dimension of the array berr must be at least max (1,nrhs)max(1,nrhs).
If INFO = 0INFO=0 or n + 1n+1, an estimate of the component-wise relative backward error of each computed solution vector jx^j (i.e., the smallest relative change in any element of AA or BB that makes jx^j an exact solution).
7:     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: fact, 2: uplo, 3: n, 4: nrhs_p, 5: a, 6: lda, 7: af, 8: ldaf, 9: ipiv, 10: b, 11: ldb, 12: x, 13: ldx, 14: rcond, 15: ferr, 16: berr, 17: work, 18: lwork, 19: rwork, 20: 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 > 0andINFONINFO>0andINFON
If infoninfon, d(i,i)d(i,i) is exactly zero. The factorization has been completed, but the factor DD is exactly singular, so the solution and error bounds could not be computed. rcond = 0.0rcond=0.0 is returned.
W INFO = N + 1INFO=N+1
DD is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of rcond would suggest.

Accuracy

For each right-hand side vector bb, the computed solution x^ is the exact solution of a perturbed system of equations (A + E) = b(A+E)x^=b, where
E1 = O(ε) A1 ,
E1 = O(ε) A1 ,
where εε is the machine precision. See Chapter 11 of Higham (2002) for further details.
If x^ is the true solution, then the computed solution xx satisfies a forward error bound of the form
( x )/( ) wc cond(A,,b)
x-x^ x^ wc cond(A,x^,b)
where cond(A,,b) = |A1|(|A||| + |b|) / cond(A) = |A1||A|κ (A) cond(A,x^,b) = |A-1| ( |A| |x^| + |b| ) / x^ cond(A) = |A-1| |A| κ (A). If x^  is the j j th column of X X , then wc wc  is returned in berr(j) berrj  and a bound on x / x - x^ / x^  is returned in ferr(j) ferrj . See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

The factorization of A A  requires approximately (4/3) n3 43 n3  floating point operations.
For each right-hand side, computation of the backward error involves a minimum of 16n2 16n2  floating point operations. Each step of iterative refinement involves an additional 24n2 24n2  operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form Ax = b Ax=b ; the number is usually 44 or 55 and never more than 1111. Each solution involves approximately 8n2 8n2  operations.
The real analogue of this function is nag_lapack_dsysvx (f07mb). The complex Hermitian analogue of this function is nag_lapack_zhesvx (f07mp).

Example

function nag_lapack_zsysvx_example
fact = 'Not factored';
uplo = 'Upper';
a = [ -0.56 + 0.12i,  -1.54 - 2.86i,  5.32 - 1.59i,  3.8 + 0.92i;
      0 + 0i,  -2.83 - 0.03i,  -3.52 + 0.58i,  -7.86 - 2.96i;
      0 + 0i,  0 + 0i,  8.86 + 1.81i,  5.14 - 0.64i;
      0 + 0i,  0 + 0i,  0 + 0i,  -0.39 - 0.71i];
af = complex(zeros(4, 4));
ipiv = zeros(4, 1, 'int64');
b = [ -6.43 + 19.24i,  -4.59 - 35.53i;
      -0.49 - 1.47i,  6.95 + 20.49i;
      -48.18 + 66i,  -12.08 - 27.02i;
      -55.64 + 41.22i,  -19.09 - 35.97i];
[afOut, ipivOut, x, rcond, ferr, berr, info] = nag_lapack_zsysvx(fact, uplo, a, af, ipiv, b)
 

afOut =

  -2.0954 - 2.2011i  -0.1071 - 0.3157i  -0.4823 + 0.0150i   0.4426 + 0.1936i
   0.0000 + 0.0000i   4.4079 + 5.3991i  -0.6078 + 0.2811i   0.5279 - 0.3715i
   0.0000 + 0.0000i   0.0000 + 0.0000i  -2.8300 - 0.0300i  -7.8600 - 2.9600i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i  -0.3900 - 0.7100i


ipivOut =

                    1
                    2
                   -2
                   -2


x =

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


rcond =

    0.0486


ferr =

   1.0e-13 *

    0.1194
    0.1181


berr =

   1.0e-16 *

    0.6383
    0.5361


info =

                    0


function f07np_example
fact = 'Not factored';
uplo = 'Upper';
a = [ -0.56 + 0.12i,  -1.54 - 2.86i,  5.32 - 1.59i,  3.8 + 0.92i;
      0 + 0i,  -2.83 - 0.03i,  -3.52 + 0.58i,  -7.86 - 2.96i;
      0 + 0i,  0 + 0i,  8.86 + 1.81i,  5.14 - 0.64i;
      0 + 0i,  0 + 0i,  0 + 0i,  -0.39 - 0.71i];
af = complex(zeros(4, 4));
ipiv = zeros(4, 1, 'int64');
b = [ -6.43 + 19.24i,  -4.59 - 35.53i;
      -0.49 - 1.47i,  6.95 + 20.49i;
      -48.18 + 66i,  -12.08 - 27.02i;
      -55.64 + 41.22i,  -19.09 - 35.97i];
[afOut, ipivOut, x, rcond, ferr, berr, info] = f07np(fact, uplo, a, af, ipiv, b)
 

afOut =

  -2.0954 - 2.2011i  -0.1071 - 0.3157i  -0.4823 + 0.0150i   0.4426 + 0.1936i
   0.0000 + 0.0000i   4.4079 + 5.3991i  -0.6078 + 0.2811i   0.5279 - 0.3715i
   0.0000 + 0.0000i   0.0000 + 0.0000i  -2.8300 - 0.0300i  -7.8600 - 2.9600i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i  -0.3900 - 0.7100i


ipivOut =

                    1
                    2
                   -2
                   -2


x =

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


rcond =

    0.0486


ferr =

   1.0e-13 *

    0.1194
    0.1181


berr =

   1.0e-16 *

    0.6383
    0.5361


info =

                    0



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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013