--- 
:name: zgelsd
:md5sum: 3cda6a4e059bdc95d6c1841204c3d144
:category: :subroutine
:arguments: 
- m: 
    :type: integer
    :intent: input
- n: 
    :type: integer
    :intent: input
- nrhs: 
    :type: integer
    :intent: input
- a: 
    :type: doublecomplex
    :intent: input
    :dims: 
    - lda
    - n
- lda: 
    :type: integer
    :intent: input
- b: 
    :type: doublecomplex
    :intent: input/output
    :dims: 
    - m
    - nrhs
    :outdims:
    - n
    - nrhs
- ldb: 
    :type: integer
    :intent: input
- s: 
    :type: doublereal
    :intent: output
    :dims: 
    - MIN(m,n)
- rcond: 
    :type: doublereal
    :intent: input
- rank: 
    :type: integer
    :intent: output
- work: 
    :type: doublecomplex
    :intent: output
    :dims: 
    - MAX(1,lwork)
- lwork: 
    :type: integer
    :intent: input
    :option: true
    :default: "m>=n ? 2*n+n*nrhs : 2*m+m*nrhs"
- rwork: 
    :type: doublereal
    :intent: workspace
    :dims: 
    - MAX(1,lrwork)
- iwork: 
    :type: integer
    :intent: workspace
    :dims: 
    - MAX(1,liwork)
- info: 
    :type: integer
    :intent: output
:substitutions: 
  m: lda
  ldb: MAX(m,n)
  c__9: "9"
  c__0: "0"
  liwork: MAX(1,3*(MIN(m,n))*nlvl+11*(MIN(m,n)))
  lrwork: "m>=n ? 10*n+2*n*smlsiz+8*n*nlvl+3*smlsiz*nrhs+(smlsiz+1)*(smlsiz+1) : 10*m+2*m*smlsiz+8*m*nlvl+2*smlsiz*nrhs+(smlsiz+1)*(smlsiz+1)"
  nlvl: MAX(0,(int)(log(1.0*MIN(m,n)/(smlsiz+1))/log(2.0)))
  smlsiz: ilaenv_(&c__9,"ZGELSD"," ",&c__0,&c__0,&c__0,&c__0)
:fortran_help: "      SUBROUTINE ZGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, RWORK, IWORK, INFO )\n\n\
  *  Purpose\n\
  *  =======\n\
  *\n\
  *  ZGELSD computes the minimum-norm solution to a real linear least\n\
  *  squares problem:\n\
  *      minimize 2-norm(| b - A*x |)\n\
  *  using the singular value decomposition (SVD) of A. A is an M-by-N\n\
  *  matrix which may be rank-deficient.\n\
  *\n\
  *  Several right hand side vectors b and solution vectors x can be\n\
  *  handled in a single call; they are stored as the columns of the\n\
  *  M-by-NRHS right hand side matrix B and the N-by-NRHS solution\n\
  *  matrix X.\n\
  *\n\
  *  The problem is solved in three steps:\n\
  *  (1) Reduce the coefficient matrix A to bidiagonal form with\n\
  *      Householder tranformations, reducing the original problem\n\
  *      into a \"bidiagonal least squares problem\" (BLS)\n\
  *  (2) Solve the BLS using a divide and conquer approach.\n\
  *  (3) Apply back all the Householder tranformations to solve\n\
  *      the original least squares problem.\n\
  *\n\
  *  The effective rank of A is determined by treating as zero those\n\
  *  singular values which are less than RCOND times the largest singular\n\
  *  value.\n\
  *\n\
  *  The divide and conquer algorithm makes very mild assumptions about\n\
  *  floating point arithmetic. It will work on machines with a guard\n\
  *  digit in add/subtract, or on those binary machines without guard\n\
  *  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or\n\
  *  Cray-2. It could conceivably fail on hexadecimal or decimal machines\n\
  *  without guard digits, but we know of none.\n\
  *\n\n\
  *  Arguments\n\
  *  =========\n\
  *\n\
  *  M       (input) INTEGER\n\
  *          The number of rows of the matrix A. M >= 0.\n\
  *\n\
  *  N       (input) INTEGER\n\
  *          The number of columns of the matrix A. N >= 0.\n\
  *\n\
  *  NRHS    (input) INTEGER\n\
  *          The number of right hand sides, i.e., the number of columns\n\
  *          of the matrices B and X. NRHS >= 0.\n\
  *\n\
  *  A       (input) COMPLEX*16 array, dimension (LDA,N)\n\
  *          On entry, the M-by-N matrix A.\n\
  *          On exit, A has been destroyed.\n\
  *\n\
  *  LDA     (input) INTEGER\n\
  *          The leading dimension of the array A. LDA >= max(1,M).\n\
  *\n\
  *  B       (input/output) COMPLEX*16 array, dimension (LDB,NRHS)\n\
  *          On entry, the M-by-NRHS right hand side matrix B.\n\
  *          On exit, B is overwritten by the N-by-NRHS solution matrix X.\n\
  *          If m >= n and RANK = n, the residual sum-of-squares for\n\
  *          the solution in the i-th column is given by the sum of\n\
  *          squares of the modulus of elements n+1:m in that column.\n\
  *\n\
  *  LDB     (input) INTEGER\n\
  *          The leading dimension of the array B.  LDB >= max(1,M,N).\n\
  *\n\
  *  S       (output) DOUBLE PRECISION array, dimension (min(M,N))\n\
  *          The singular values of A in decreasing order.\n\
  *          The condition number of A in the 2-norm = S(1)/S(min(m,n)).\n\
  *\n\
  *  RCOND   (input) DOUBLE PRECISION\n\
  *          RCOND is used to determine the effective rank of A.\n\
  *          Singular values S(i) <= RCOND*S(1) are treated as zero.\n\
  *          If RCOND < 0, machine precision is used instead.\n\
  *\n\
  *  RANK    (output) INTEGER\n\
  *          The effective rank of A, i.e., the number of singular values\n\
  *          which are greater than RCOND*S(1).\n\
  *\n\
  *  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))\n\
  *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.\n\
  *\n\
  *  LWORK   (input) INTEGER\n\
  *          The dimension of the array WORK. LWORK must be at least 1.\n\
  *          The exact minimum amount of workspace needed depends on M,\n\
  *          N and NRHS. As long as LWORK is at least\n\
  *              2*N + N*NRHS\n\
  *          if M is greater than or equal to N or\n\
  *              2*M + M*NRHS\n\
  *          if M is less than N, the code will execute correctly.\n\
  *          For good performance, LWORK should generally be larger.\n\
  *\n\
  *          If LWORK = -1, then a workspace query is assumed; the routine\n\
  *          only calculates the optimal size of the array WORK and the\n\
  *          minimum sizes of the arrays RWORK and IWORK, and returns\n\
  *          these values as the first entries of the WORK, RWORK and\n\
  *          IWORK arrays, and no error message related to LWORK is issued\n\
  *          by XERBLA.\n\
  *\n\
  *  RWORK   (workspace) DOUBLE PRECISION array, dimension (MAX(1,LRWORK))\n\
  *          LRWORK >=\n\
  *             10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +\n\
  *             MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )\n\
  *          if M is greater than or equal to N or\n\
  *             10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +\n\
  *             MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )\n\
  *          if M is less than N, the code will execute correctly.\n\
  *          SMLSIZ is returned by ILAENV and is equal to the maximum\n\
  *          size of the subproblems at the bottom of the computation\n\
  *          tree (usually about 25), and\n\
  *             NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )\n\
  *          On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK.\n\
  *\n\
  *  IWORK   (workspace) INTEGER array, dimension (MAX(1,LIWORK))\n\
  *          LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),\n\
  *          where MINMN = MIN( M,N ).\n\
  *          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.\n\
  *\n\
  *  INFO    (output) INTEGER\n\
  *          = 0: successful exit\n\
  *          < 0: if INFO = -i, the i-th argument had an illegal value.\n\
  *          > 0:  the algorithm for computing the SVD failed to converge;\n\
  *                if INFO = i, i off-diagonal elements of an intermediate\n\
  *                bidiagonal form did not converge to zero.\n\
  *\n\n\
  *  Further Details\n\
  *  ===============\n\
  *\n\
  *  Based on contributions by\n\
  *     Ming Gu and Ren-Cang Li, Computer Science Division, University of\n\
  *       California at Berkeley, USA\n\
  *     Osni Marques, LBNL/NERSC, USA\n\
  *\n\
  *  =====================================================================\n\
  *\n"
