CLALSD uses the singular value decomposition of A to solve the least


 squares problem of finding X to minimize the Euclidean norm of each


 column of A*X-B, where A is N-by-N upper bidiagonal, and X and B


 are N-by-NRHS. The solution X overwrites B.


 The singular values of A smaller than RCOND times the largest


 singular value are treated as zero in solving the least squares


 problem; in this case a minimum norm solution is returned.


 The actual singular values are returned in D in ascending order.


 This code makes very mild assumptions about floating point


 arithmetic. It will work on machines with a guard digit in


 add/subtract, or on those binary machines without guard digits


 which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.


 It could conceivably fail on hexadecimal or decimal machines


 without guard digits, but we know of none.

UPLO

          UPLO is CHARACTER*1


         = 'U': D and E define an upper bidiagonal matrix.


         = 'L': D and E define a  lower bidiagonal matrix.

SMLSIZ

          SMLSIZ is INTEGER


         The maximum size of the subproblems at the bottom of the


         computation tree.

N

          N is INTEGER


         The dimension of the  bidiagonal matrix.  N >= 0.

NRHS

          NRHS is INTEGER


         The number of columns of B. NRHS must be at least 1.

D

          D is REAL array, dimension (N)


         On entry D contains the main diagonal of the bidiagonal


         matrix. On exit, if INFO = 0, D contains its singular values.

E

          E is REAL array, dimension (N-1)


         Contains the super-diagonal entries of the bidiagonal matrix.


         On exit, E has been destroyed.

B

          B is COMPLEX array, dimension (LDB,NRHS)


         On input, B contains the right hand sides of the least


         squares problem. On output, B contains the solution X.

LDB

          LDB is INTEGER


         The leading dimension of B in the calling subprogram.


         LDB must be at least max(1,N).

RCOND

          RCOND is REAL


         The singular values of A less than or equal to RCOND times


         the largest singular value are treated as zero in solving


         the least squares problem. If RCOND is negative,


         machine precision is used instead.


         For example, if diag(S)*X=B were the least squares problem,


         where diag(S) is a diagonal matrix of singular values, the


         solution would be X(i) = B(i) / S(i) if S(i) is greater than


         RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to


         RCOND*max(S).

RANK

          RANK is INTEGER


         The number of singular values of A greater than RCOND times


         the largest singular value.

WORK

          WORK is COMPLEX array, dimension (N * NRHS).

RWORK

          RWORK is REAL array, dimension at least


         (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +


         MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ),


         where


         NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )

IWORK

          IWORK is INTEGER array, dimension (3*N*NLVL + 11*N).

INFO

          INFO is INTEGER


         = 0:  successful exit.


         < 0:  if INFO = -i, the i-th argument had an illegal value.


         > 0:  The algorithm failed to compute a singular value while


               working on the submatrix lying in rows and columns


               INFO/(N+1) through MOD(INFO,N+1).

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

September 2012

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA