DGELSD computes the minimum-norm solution to a real linear least


 squares problem:


     minimize 2-norm(| b - A*x |)


 using the singular value decomposition (SVD) of A. A is an M-by-N


 matrix which may be rank-deficient.


 Several right hand side vectors b and solution vectors x can be


 handled in a single call; they are stored as the columns of the


 M-by-NRHS right hand side matrix B and the N-by-NRHS solution


 matrix X.


 The problem is solved in three steps:


 (1) Reduce the coefficient matrix A 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 tranformations to solve


     the original least squares problem.


 The effective rank of A is determined by treating as zero those


 singular values which are less than RCOND times the largest singular


 value.


 The divide and conquer algorithm 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.

M

          M is INTEGER


          The number of rows of A. M >= 0.

N

          N is INTEGER


          The number of columns of A. N >= 0.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrices B and X. NRHS >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)


          On entry, the M-by-N matrix A.


          On exit, A has been destroyed.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,M).

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)


          On entry, the M-by-NRHS right hand side matrix B.


          On exit, B is overwritten by the N-by-NRHS solution


          matrix X.  If m >= n and RANK = n, the residual


          sum-of-squares for the solution in the i-th column is given


          by the sum of squares of elements n+1:m in that column.

LDB

          LDB is INTEGER


          The leading dimension of the array B. LDB >= max(1,max(M,N)).

S

          S is DOUBLE PRECISION array, dimension (min(M,N))


          The singular values of A in decreasing order.


          The condition number of A in the 2-norm = S(1)/S(min(m,n)).

RCOND

          RCOND is DOUBLE PRECISION


          RCOND is used to determine the effective rank of A.


          Singular values S(i) <= RCOND*S(1) are treated as zero.


          If RCOND < 0, machine precision is used instead.

RANK

          RANK is INTEGER


          The effective rank of A, i.e., the number of singular values


          which are greater than RCOND*S(1).

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK. LWORK must be at least 1.


          The exact minimum amount of workspace needed depends on M,


          N and NRHS. As long as LWORK is at least


              12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,


          if M is greater than or equal to N or


              12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,


          if M is less than N, the code will execute correctly.


          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), and


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


          For good performance, LWORK should generally be larger.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (MAX(1,LIWORK))


          LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN),


          where MINMN = MIN( M,N ).


          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

INFO

          INFO is INTEGER


          = 0:  successful exit


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


          > 0:  the algorithm for computing the SVD failed to converge;


                if INFO = i, i off-diagonal elements of an intermediate


                bidiagonal form did not converge to zero.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011

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