SBDSDC computes the singular value decomposition (SVD) of a real


 N-by-N (upper or lower) bidiagonal matrix B:  B = U * S * VT,


 using a divide and conquer method, where S is a diagonal matrix


 with non-negative diagonal elements (the singular values of B), and


 U and VT are orthogonal matrices of left and right singular vectors,


 respectively. SBDSDC can be used to compute all singular values,


 and optionally, singular vectors or singular vectors in compact form.


 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 X-MP, Cray Y-MP, Cray C-90, or Cray-2.


 It could conceivably fail on hexadecimal or decimal machines


 without guard digits, but we know of none.  See SLASD3 for details.


 The code currently calls SLASDQ if singular values only are desired.


 However, it can be slightly modified to compute singular values


 using the divide and conquer method.

UPLO

          UPLO is CHARACTER*1


          = 'U':  B is upper bidiagonal.


          = 'L':  B is lower bidiagonal.

COMPQ

          COMPQ is CHARACTER*1


          Specifies whether singular vectors are to be computed


          as follows:


          = 'N':  Compute singular values only;


          = 'P':  Compute singular values and compute singular


                  vectors in compact form;


          = 'I':  Compute singular values and singular vectors.

N

          N is INTEGER


          The order of the matrix B.  N >= 0.

D

          D is REAL array, dimension (N)


          On entry, the n diagonal elements of the bidiagonal matrix B.


          On exit, if INFO=0, the singular values of B.

E

          E is REAL array, dimension (N-1)


          On entry, the elements of E contain the offdiagonal


          elements of the bidiagonal matrix whose SVD is desired.


          On exit, E has been destroyed.

U

          U is REAL array, dimension (LDU,N)


          If  COMPQ = 'I', then:


             On exit, if INFO = 0, U contains the left singular vectors


             of the bidiagonal matrix.


          For other values of COMPQ, U is not referenced.

LDU

          LDU is INTEGER


          The leading dimension of the array U.  LDU >= 1.


          If singular vectors are desired, then LDU >= max( 1, N ).

VT

          VT is REAL array, dimension (LDVT,N)


          If  COMPQ = 'I', then:


             On exit, if INFO = 0, VT**T contains the right singular


             vectors of the bidiagonal matrix.


          For other values of COMPQ, VT is not referenced.

LDVT

          LDVT is INTEGER


          The leading dimension of the array VT.  LDVT >= 1.


          If singular vectors are desired, then LDVT >= max( 1, N ).

Q

          Q is REAL array, dimension (LDQ)


          If  COMPQ = 'P', then:


             On exit, if INFO = 0, Q and IQ contain the left


             and right singular vectors in a compact form,


             requiring O(N log N) space instead of 2*N**2.


             In particular, Q contains all the REAL data in


             LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1))))


             words of memory, where SMLSIZ is returned by ILAENV and


             is equal to the maximum size of the subproblems at the


             bottom of the computation tree (usually about 25).


          For other values of COMPQ, Q is not referenced.

IQ

          IQ is INTEGER array, dimension (LDIQ)


          If  COMPQ = 'P', then:


             On exit, if INFO = 0, Q and IQ contain the left


             and right singular vectors in a compact form,


             requiring O(N log N) space instead of 2*N**2.


             In particular, IQ contains all INTEGER data in


             LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))


             words of memory, where SMLSIZ is returned by ILAENV and


             is equal to the maximum size of the subproblems at the


             bottom of the computation tree (usually about 25).


          For other values of COMPQ, IQ is not referenced.

WORK

          WORK is REAL array, dimension (MAX(1,LWORK))


          If COMPQ = 'N' then LWORK >= (4 * N).


          If COMPQ = 'P' then LWORK >= (6 * N).


          If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).

IWORK

          IWORK is INTEGER array, dimension (8*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.


                The update process of divide and conquer failed.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA