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NAG Toolbox

NAG Toolbox: nag_lapack_zgelsd (f08kq)

Purpose

nag_lapack_zgelsd (f08kq) computes the minimum norm solution to a complex linear least squares problem
min bAx2.
x
minx b-Ax2 .

Syntax

[a, b, s, rank, info] = f08kq(a, b, rcond, 'm', m, 'n', n, 'nrhs_p', nrhs_p, 'lwork', lwork)
[a, b, s, rank, info] = nag_lapack_zgelsd(a, b, rcond, 'm', m, 'n', n, 'nrhs_p', nrhs_p, 'lwork', lwork)

Description

nag_lapack_zgelsd (f08kq) uses the singular value decomposition (SVD) of AA, where AA is a complex mm by nn matrix which may be rank-deficient.
Several right-hand side vectors bb and solution vectors xx can be handled in a single call; they are stored as the columns of the mm by rr right-hand side matrix BB and the nn by rr solution matrix XX.
The problem is solved in three steps:
  1. reduce the coefficient matrix AA to bidiagonal form with Householder transformations, reducing the original problem into a ‘bidiagonal least squares problem’ (BLS);
  2. solve the BLS using a divide-and-conquer approach;
  3. apply back all the Householder transformations to solve the original least squares problem.
The effective rank of AA is determined by treating as zero those singular values which are less than rcond times the largest singular value.

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:     a(lda, : :) – complex array
The first dimension of the array a must be at least max (1,m)max(1,m)
The second dimension of the array must be at least max (1,n)max(1,n)
The mm by nn coefficient matrix AA.
2:     b(ldb, : :) – complex array
The first dimension of the array b must be at least max (1,m,n)max(1,m,n)
The second dimension of the array must be at least max (1,nrhs)max(1,nrhs)
The mm by rr right-hand side matrix BB.
3:     rcond – double scalar
Used to determine the effective rank of AA. Singular values s(i)rcond × s(1)sircond×s1 are treated as zero. If rcond < 0rcond<0, machine precision is used instead.

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the array a.
mm, the number of rows of the matrix AA.
Constraint: m0m0.
2:     n – int64int32nag_int scalar
Default: The second dimension of the array a.
nn, the number of columns of the matrix AA.
Constraint: n0n0.
3:     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 matrices BB and XX.
Constraint: nrhs0nrhs0.
4:     lwork – int64int32nag_int scalar
The dimension of the array work as declared in the (sub)program from which nag_lapack_zgelsd (f08kq) is called.
The exact minimum amount of workspace needed depends on m, n and nrhs_p. As long as lwork is at least
max (1,m + n + r,2r + r × nrhs) ,
max(1,m+n+r,2r+r×nrhs) ,
where r = min (m,n)r=min(m,n), the code will execute correctly.
Suggested value: for optimal performance, lwork should generally be larger than the required minimum. Consider increasing lwork by at least nb × min (m,n) nb×min(m,n) , where nb nb  is the optimal block size.
Default: max (1, 64 min (m,n) max (m + n + r,2r + r × nrhs) ) max(1, 64 min(m,n) max(m+n+r,2r+r×nrhs) )  
Constraint: lwork must be at least max (1,m + n + r,2r + r × nrhs)max(1,m+n+r,2r+r×nrhs).

Input Parameters Omitted from the MATLAB Interface

lda ldb work rwork iwork

Output Parameters

1:     a(lda, : :) – complex array
The first dimension of the array a will be max (1,m)max(1,m)
The second dimension of the array will be max (1,n)max(1,n)
ldamax (1,m)ldamax(1,m).
The contents of a are destroyed.
2:     b(ldb, : :) – complex array
The first dimension of the array b will be max (1,m,n)max(1,m,n)
The second dimension of the array will be max (1,nrhs)max(1,nrhs)
ldbmax (1,m,n)ldbmax(1,m,n).
b stores the nn by rr solution matrix XX. If mnmn and rank = nrank=n, the residual sum of squares for the solution in the iith column is given by the sum of squares of the modulus of elements n + 1,,mn+1,,m in that column.
3:     s( : :) – double array
Note: the dimension of the array s must be at least max (1,min (m,n)) max(1,min(m,n)) .
The singular values of AA in decreasing order.
4:     rank – int64int32nag_int scalar
The effective rank of AA, i.e., the number of singular values which are greater than rcond × s(1)rcond×s1.
5:     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: m, 2: n, 3: nrhs_p, 4: a, 5: lda, 6: b, 7: ldb, 8: s, 9: rcond, 10: rank, 11: work, 12: lwork, 13: rwork, 14: iwork, 15: 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.
  INFO > 0INFO>0
The algorithm for computing the SVD failed to converge; if info = iinfo=i, ii off-diagonal elements of an intermediate bidiagonal form did not converge to zero.

Accuracy

See Section 4.5 of Anderson et al. (1999) for details.

Further Comments

The real analogue of this function is nag_lapack_dgelsd (f08kc).

Example

function nag_lapack_zgelsd_example
a = [ 0.47 - 0.34i,  -0.32 - 0.23i,  0.35 - 0.6i,  0.89 + 0.71i,  -0.19 + 0.06i;
      -0.4 + 0.54i,  -0.05 + 0.2i,  -0.52 - 0.34i, ...
      -0.45 - 0.45i,  0.11 - 0.85i;
      0.6 + 0.01i,  -0.26 - 0.44i,  0.87 - 0.11i,  -0.02 - 0.57i,  1.44 + 0.8i;
      0.8 - 1.02i,  -0.43 + 0.17i,  -0.34 - 0.09i, ...
      1.14 - 0.78i,  0.07 + 1.14i];
b = [ 2.15 - 0.2i;
      -2.24 + 1.82i;
      4.45 - 4.28i;
      5.7 - 6.25i;
      0 + 0i];
rcond = 0.01;
[aOut, bOut, s, rank, info] = nag_lapack_zgelsd(a, b, rcond)
 

aOut =

  -1.5199 + 0.0000i  -0.1371 - 0.1390i   0.2210 - 0.2638i   0.3753 + 0.4209i  -0.0978 + 0.0134i
  -1.9194 + 0.0000i  -1.4211 + 0.0000i  -0.2911 + 0.4160i   0.0463 - 0.4369i  -0.2470 + 0.5898i
  -0.0459 - 0.0067i   1.9985 + 0.0000i   0.9249 + 0.0000i   0.3343 + 0.4472i  -0.1840 + 0.1355i
  -0.2659 + 0.5880i   0.1704 + 0.3519i  -1.0610 + 0.0000i   0.0192 + 0.0000i   0.3393 - 0.6232i


bOut =

   3.9747 - 1.8377i
  -0.9186 + 0.8253i
  -0.3105 + 0.1477i
   1.0050 + 0.8626i
  -0.2256 - 1.9425i


s =

    2.9979
    1.9983
    1.0044
    0.0064


rank =

                    3


info =

                    0


function f08kq_example
a = [ 0.47 - 0.34i,  -0.32 - 0.23i,  0.35 - 0.6i,  0.89 + 0.71i,  -0.19 + 0.06i;
      -0.4 + 0.54i,  -0.05 + 0.2i,  -0.52 - 0.34i, ...
      -0.45 - 0.45i,  0.11 - 0.85i;
      0.6 + 0.01i,  -0.26 - 0.44i,  0.87 - 0.11i,  -0.02 - 0.57i,  1.44 + 0.8i;
      0.8 - 1.02i,  -0.43 + 0.17i,  -0.34 - 0.09i, ...
      1.14 - 0.78i,  0.07 + 1.14i];
b = [ 2.15 - 0.2i;
      -2.24 + 1.82i;
      4.45 - 4.28i;
      5.7 - 6.25i;
      0 + 0i];
rcond = 0.01;
[aOut, bOut, s, rank, info] = f08kq(a, b, rcond)
 

aOut =

  -1.5199 + 0.0000i  -0.1371 - 0.1390i   0.2210 - 0.2638i   0.3753 + 0.4209i  -0.0978 + 0.0134i
  -1.9194 + 0.0000i  -1.4211 + 0.0000i  -0.2911 + 0.4160i   0.0463 - 0.4369i  -0.2470 + 0.5898i
  -0.0459 - 0.0067i   1.9985 + 0.0000i   0.9249 + 0.0000i   0.3343 + 0.4472i  -0.1840 + 0.1355i
  -0.2659 + 0.5880i   0.1704 + 0.3519i  -1.0610 + 0.0000i   0.0192 + 0.0000i   0.3393 - 0.6232i


bOut =

   3.9747 - 1.8377i
  -0.9186 + 0.8253i
  -0.3105 + 0.1477i
   1.0050 + 0.8626i
  -0.2256 - 1.9425i


s =

    2.9979
    1.9983
    1.0044
    0.0064


rank =

                    3


info =

                    0



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