From Mark Gates (UTK) - Fix Bug 157 - Various bugs in SVD routines (*gesdd, *gesvd, and *bdsdc).

Items are labelled (a) through (m), omitting (l).
Several are not bugs, just suggestions.

Most bugs are in *gesdd.

There's one bug (g) in *bdsdc. This is the underlying cause of LAPACK bug #111.

There's one bug (m) in [cz]gesvd. I also added an INT() cast in these
assignments to silence compiler warnings. Changed:
    LWORK_ZGEQRF=CDUM(1)
to:
    LWORK_ZGEQRF = INT( CDUM(1) )

Where possible, I ran a test showing the wrong behavior, then a test showing the
corrected behavior. These use a modified version of the MAGMA SVD tester
(calling LAPACK), because I could adjust the lwork as needed. The last 3 columns
are the lwork type, the lwork size, and the lwork formula. The lwork types are:

    doc_old  as documented in LAPACK 3.6.
    doc      as in the attached, updated documentation.
    min_old  minwrk, as computed in LAPACK 3.6.
    min      minwrk, as computed in the attached, updated code.
    min-1    minimum - 1; this should cause gesdd to return an error.
    opt      optimal size.
    max      the maximum size LAPACK will take advantage of;
             some cases, the optimal is n*n + work, while the max is m*n + work.
    query    what gesdd returns for an lwork query; should equal opt or max.

After the lwork, occasionally there is a ! or ? error code indicating:
Error codes:  !  error:  lwork < min. For (min-1), this ought to appear.
              ?  compatability issue:  lwork < min_old, will fail for lapack <= 3.6.

I also tested the update routines on a wide variety of sizes and jobz, with
various lwork.

Besides fixing the bugs below, I made two significant changes.

1)  Changed *gesdd from computing each routine's workspace using, e.g.:
         N*ilaenv(...)
    to querying each routine for its LWORK, e.g.:
                CALL ZGEBRD( M, N, CDUM(1), M, CDUM(1), DUM(1), CDUM(1),
         $                   CDUM(1), CDUM(1), -1, IERR )
                LWORK_ZGEBRD_MN = INT( CDUM(1) )
        
    This matches how *gesvd was changed in LAPACK 3.4.0.

2)  Changed the Workspace: comments, which were incredibly deceptive.
    For instance, in Path 2 before dbdsdc, it said
        Workspace: need N + N*N + BDSPAC
    since dbdsdc needs the [e] vector, [U] matrix, and bdspac.
    However, that ignores that the [tauq, taup] vectors and [R] matrix
    are also already allocated, though dbdsdc doesn't need them.
    So the workspace needed at that point in the code is actually
        Workspace: need N*N [R] + 3*N [e, tauq, taup] + N*N [U] + BDSPAC
    For clarity, I added in [brackets] what matrices or vectors were allocated,
    and the order reflects their order in memory.
    
    I may do a similar change for *gesvd eventually. The workspace comments in
    MAGMA's *gesvd have already been updated as above.



================================================================================
a)  Throughout, to simplify equations, let:

    mn = min( M, N )
    mx = max( M, N )



================================================================================
b)  [sdcz]gesdd Path 4 (m >> n, job="all") has wrong minwrk formula in code:
        minwrk = bdspac + mn*mn + 2*mn + mx
               = 4*mn*mn + 6*mn + mx
    This is an overestimate, needlessly rejecting the documented formula:
        doc    = 4*mn*mn + 7*mn
    In complex, the correct min fails, but the doc matches the wrong minwrk.
    Solution: fix code to:
        minwrk = mn*mn + max( 3*mn + bdspac, mn + mx )
               = mn*mn + max( 3*mn*mn + 7*mn, mn + mx )

     Test cases:
        m=40, ..., 100;  n=20;  jobz='A'   
     
    See bug (d) showing documentation is also wrong.
    Also, see bug (i), complex [cz]gesdd should return -12 instead of -13.

================================================================================
bt) transposed case
    [sd]gesdd Path 4t (n >> m, job="all") has a different wrong minwrk; see bug (c).
    [cz]gesdd Path 4t exhibits same bug as Path 4.

    Test cases:
        m=20;  n=40, ..., 100;  jobz='A'

================================================================================
c)  [sd]gesdd Path 4t (n >> m, job="all") has wrong minwrk formula in code,
    different than bug (b):
        minwrk = bdspac + m*m + 3*m
               = 4*mn*mn + 7*mn
    This formula lacks any dependence on N, so the code will fail (without
    setting info; orglq calls xerbla) if N is too large, N > 3*M*M + 6*M.
    Bug does not occur in complex.
    
    Test cases:
        m=20;  n = 1320;  jobz='A'  ok    with documented lwork
        m=20;  n > 1320;  jobz='A'  fails with documented lwork
    
    Solution: as in bug (b), fix code to:
        minwrk = mn*mn + max( 3*mn + bdspac, mn + mx )
               = mn*mn + max( 3*mn*mn + 7*mn, mn + mx )
    
    See bug (d) showing documentation is also wrong.

================================================================================
d)  [sd]gesdd documentation lists the same minimum size for jobz='S' and 'A':
        If JOBZ = 'S' or 'A', LWORK >= min(M,N)*(7 + 4*min(M,N))
    However, jobz='A' actually also depends on max(M,N):
        minwrk = mn*mn + max( 3*mn*mn + 7*mn, mn + mx )
    This causes the formula to fail for mx > 3*mn*mn + 6*mn.
    
    Test cases:
        m > 1320;  n = 20;    jobz='A'  fails with document lwork, even after fixing bugs (b) and (c).
        m = 20;    n > 1320;  jobz='A'  fails also.
    
    Solution: in docs, split these two cases. This fix uses an overestimate,
    so that codes using it will be backwards compatible with LAPACK <= 3.6.

        If JOBZ = 'S', LWORK >= 4*mn*mn + 7*mn.
        If JOBZ = 'A', LWORK >= 4*mn*mn + 6*mn + mx.

================================================================================
e)  [sd]gesdd, Path 5, jobz='A' has wrong maxwrk formula in the code:
        MAXWRK = MAX( MAXWRK, BDSPAC + 3*N )
    Should be:
        MAXWRK = MAX( WRKBL, BDSPAC + 3*N )
    This causes the lwork query to ignore WRKBL, and return the minimum
    workspace size, BDSPAC + 3*N, instead of the optimal workspace size.
    However, it only affects the result for small sizes where min(M,N) < NB.
    Path 5t has the correct maxwrk formula.
    Complex is correct for both Path 5 and 5t.
    
    Test case:
        Compare lwork query with
        M = 30, N = 20, jobz='A', lwork query is 1340
        M = 20, N = 30, jobz='A', lwork query is 3260
        These should be the same.

    Solution: fix code as above.

================================================================================
f)  Not a bug, just a suggestion.
    The lwork minimum sizes are not actually minimums, and can be larger than
    the queried lwork size.
    
    Solution: add a comment:
    These are not tight minimums in all cases; see comments inside code.

================================================================================
g)  [sd]bdsdc segfaults due to too small workspace size. Its documentation claims:
        If COMPQ = 'N' then LWORK >= (4 * N).
    Based on this, in zgesdd, the rwork size >= 5*min(M,N).
    However, LAPACK bug 111 found that rwork size >= 7*min(M,N) was required.

    In dbdsdc, if uplo='L', then it rotates lower bidiagonal to upper bidiagonal,
    and saves 2 vectors of Givens rotations in work. It shifts WSTART from
    1 to 2*N-1. Then it calls dlasdq( ..., work( wstart ), info ).
    As dlasdq requires 4*N, dbdsdc would now require 6*N in this case.
    This caused zgesdd to require rwork size >= 7*min(M,N) when N > M and jobz='N'.

    My preferred solution is to change WSTART to 1 in the DLASDQ call inside dbdsdc:

          IF( ICOMPQ.EQ.0 ) THEN
             CALL DLASDQ( 'U', 0, N, 0, 0, 0, D, E, VT, LDVT, U, LDU, U,
         $                LDU, WORK( WSTART ), INFO )
             GO TO 40
          END IF

    to:

          IF( ICOMPQ.EQ.0 ) THEN
    *        Ignores WSTART, which is needed only for ICOMPQ = 1 or 2;
    *        using WSTART would change required workspace to 6*N for uplo='L'.
             CALL DLASDQ( 'U', 0, N, 0, 0, 0, D, E, VT, LDVT, U, LDU, U,
         $                LDU, WORK( 1 ), INFO )
             GO TO 40
          END IF

    The [cz]gesdd documentation, which was changed to 7*min(M,N) in LAPACK 3.6,
    may be reverted to 5*min(M,N), if desired.

================================================================================
h)  [sd]gesdd for jobz='N' requires bdspac = 7*n for the dbdsdc workspace.
    However, dbdsdc requires only 4*n, or 6*n before fixing bug (g).
    For backwards compatability, I did not change the code, but added a comment
    for clarification.


================================================================================
i)  [cz]gesdd returns info = -13 instead of info = -12 for lwork errors.


================================================================================
j)  In zgesdd, for computing maxwrk, these paths:
        Path 6,  jobz=A
        Path 6t, jobz=S
        Path 6t, jobz=A
    query ilaenv( 1, zungbr, ... )
    when the code actually calls zunmbr (twice). I corrected it.


================================================================================
k)  In zgesdd documentation, currently
        lrwork >= max( 5*mn*mn + 7*mn, 2*mx*mn + 2*mn*mn + mn )

    It doesn't need that much, particularly for (mx >> mn) case.
        If (mx >> mn), lrwork >= 5*mn*mn + 5*mn;
        else,          lrwork >= max( 5*mn*mn + 5*mn,
                                      2*mx*mn + 2*mn*mn + mn ).

    I changed this in the documentation. Feel free to revert if you prefer.



================================================================================
m)  [cz]gesvd, Path 10 and 10t, have minwrk inside the wrong conditional:
        IF( .NOT.WNTVN ) THEN
           MAXWRK = MAX( MAXWRK, 2*N+LWORK_ZUNGBR_P )
        MINWRK = 2*N + M
        END IF
    So Path 10 with jobvt='N', and Path 10t with jobu='N', have minwrk = 1,
    so an invalid lwork is not correctly rejected.

================================================================================
mt) transposed case
    broken: with old routine, Path 10t with jobu='N' doesn't enforce minwrk

1 files changed, 420 insertions, 301 deletions

diff --git a/SRC/dgesdd.f b/SRC/dgesdd.f
index 54e2652e..4bdc8a64 100644
--- a/SRC/dgesdd.f
+++ b/SRC/dgesdd.f
@@ -18,8 +18,8 @@
 *  Definition:
 *  ===========
 *
-*       SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
-*                          LWORK, IWORK, INFO )
+*       SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
+*                          WORK, LWORK, IWORK, INFO )
 * 
 *       .. Scalar Arguments ..
 *       CHARACTER          JOBZ
@@ -154,8 +154,8 @@
 *> \param[in] LDVT
 *> \verbatim
 *>          LDVT is INTEGER
-*>          The leading dimension of the array VT.  LDVT >= 1; if
-*>          JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
+*>          The leading dimension of the array VT.  LDVT >= 1;
+*>          if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
 *>          if JOBZ = 'S', LDVT >= min(M,N).
 *> \endverbatim
 *>
@@ -169,16 +169,18 @@
 *> \verbatim
 *>          LWORK is INTEGER
 *>          The dimension of the array WORK. LWORK >= 1.
-*>          If JOBZ = 'N',
-*>            LWORK >= 3*min(M,N) + max(max(M,N),7*min(M,N)).
-*>          If JOBZ = 'O',
-*>            LWORK >= 3*min(M,N) + 
-*>                     max(max(M,N),5*min(M,N)*min(M,N)+4*min(M,N)).
-*>          If JOBZ = 'S' or 'A'
-*>            LWORK >= min(M,N)*(7+4*min(M,N))
-*>          For good performance, LWORK should generally be larger.
-*>          If LWORK = -1 but other input arguments are legal, WORK(1)
-*>          returns the optimal LWORK.
+*>          If LWORK = -1, a workspace query is assumed.  The optimal
+*>          size for the WORK array is calculated and stored in WORK(1),
+*>          and no other work except argument checking is performed.
+*>
+*>          Let mx = max(M,N) and mn = min(M,N).
+*>          If JOBZ = 'N', LWORK >= 3*mn + max( mx, 7*mn ).
+*>          If JOBZ = 'O', LWORK >= 3*mn + max( mx, 5*mn*mn + 4*mn ).
+*>          If JOBZ = 'S', LWORK >= 4*mn*mn + 7*mn.
+*>          If JOBZ = 'A', LWORK >= 4*mn*mn + 6*mn + mx.
+*>          These are not tight minimums in all cases; see comments inside code.
+*>          For good performance, LWORK should generally be larger;
+*>          a query is recommended.
 *> \endverbatim
 *>
 *> \param[out] IWORK
@@ -212,9 +214,11 @@
 *>     Ming Gu and Huan Ren, Computer Science Division, University of
 *>     California at Berkeley, USA
 *>
+*> @precisions fortran d -> s
 *  =====================================================================
-      SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
-     $                   LWORK, IWORK, INFO )
+      SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
+     $                   WORK, LWORK, IWORK, INFO )
+      implicit none
 *
 *  -- LAPACK driver routine (version 3.6.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
@@ -243,6 +247,15 @@
      $                   IR, ISCL, ITAU, ITAUP, ITAUQ, IU, IVT, LDWKVT,
      $                   LDWRKL, LDWRKR, LDWRKU, MAXWRK, MINMN, MINWRK,
      $                   MNTHR, NWORK, WRKBL
+      INTEGER            LWORK_DGEBRD_MN, LWORK_DGEBRD_MM,  
+     $                   LWORK_DGEBRD_NN, LWORK_DGELQF_MN,
+     $                   LWORK_DGEQRF_MN,
+     $                   LWORK_DORGBR_P_MM, LWORK_DORGBR_Q_NN,
+     $                   LWORK_DORGLQ_MN, LWORK_DORGLQ_NN,
+     $                   LWORK_DORGQR_MM, LWORK_DORGQR_MN,
+     $                   LWORK_DORMBR_PRT_MM, LWORK_DORMBR_QLN_MM,
+     $                   LWORK_DORMBR_PRT_MN, LWORK_DORMBR_QLN_MN,
+     $                   LWORK_DORMBR_PRT_NN, LWORK_DORMBR_QLN_NN
       DOUBLE PRECISION   ANRM, BIGNUM, EPS, SMLNUM
 *     ..
 *     .. Local Arrays ..
@@ -256,9 +269,8 @@
 *     ..
 *     .. External Functions ..
       LOGICAL            LSAME
-      INTEGER            ILAENV
       DOUBLE PRECISION   DLAMCH, DLANGE
-      EXTERNAL           DLAMCH, DLANGE, ILAENV, LSAME
+      EXTERNAL           DLAMCH, DLANGE, LSAME
 *     ..
 *     .. Intrinsic Functions ..
       INTRINSIC          INT, MAX, MIN, SQRT
@@ -267,13 +279,13 @@
 *
 *     Test the input arguments
 *
-      INFO = 0
-      MINMN = MIN( M, N )
-      WNTQA = LSAME( JOBZ, 'A' )
-      WNTQS = LSAME( JOBZ, 'S' )
+      INFO   = 0
+      MINMN  = MIN( M, N )
+      WNTQA  = LSAME( JOBZ, 'A' )
+      WNTQS  = LSAME( JOBZ, 'S' )
       WNTQAS = WNTQA .OR. WNTQS
-      WNTQO = LSAME( JOBZ, 'O' )
-      WNTQN = LSAME( JOBZ, 'N' )
+      WNTQO  = LSAME( JOBZ, 'O' )
+      WNTQN  = LSAME( JOBZ, 'N' )
       LQUERY = ( LWORK.EQ.-1 )
 *
       IF( .NOT.( WNTQA .OR. WNTQS .OR. WNTQO .OR. WNTQN ) ) THEN
@@ -294,115 +306,140 @@
       END IF
 *
 *     Compute workspace
-*      (Note: Comments in the code beginning "Workspace:" describe the
-*       minimal amount of workspace needed at that point in the code,
+*       Note: Comments in the code beginning "Workspace:" describe the
+*       minimal amount of workspace allocated at that point in the code,
 *       as well as the preferred amount for good performance.
 *       NB refers to the optimal block size for the immediately
-*       following subroutine, as returned by ILAENV.)
+*       following subroutine, as returned by ILAENV.
 *
       IF( INFO.EQ.0 ) THEN
          MINWRK = 1
          MAXWRK = 1
+         BDSPAC = 0
+         MNTHR  = INT( MINMN*11.0D0 / 6.0D0 )
          IF( M.GE.N .AND. MINMN.GT.0 ) THEN
 *
 *           Compute space needed for DBDSDC
 *
-            MNTHR = INT( MINMN*11.0D0 / 6.0D0 )
             IF( WNTQN ) THEN
+*              dbdsdc needs only 4*N (or 6*N for uplo=L for LAPACK <= 3.6)
+*              keep 7*N for backwards compatability.
                BDSPAC = 7*N
             ELSE
                BDSPAC = 3*N*N + 4*N
             END IF
+*
+*           Compute space preferred for each routine
+            CALL DGEBRD( M, N, DUM(1), M, DUM(1), DUM(1), DUM(1),
+     $                   DUM(1), DUM(1), -1, IERR )
+            LWORK_DGEBRD_MN = INT( DUM(1) )
+*
+            CALL DGEBRD( N, N, DUM(1), N, DUM(1), DUM(1), DUM(1),
+     $                   DUM(1), DUM(1), -1, IERR )
+            LWORK_DGEBRD_NN = INT( DUM(1) )
+*
+            CALL DGEQRF( M, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
+            LWORK_DGEQRF_MN = INT( DUM(1) )
+*
+            CALL DORGBR( 'Q', N, N, N, DUM(1), N, DUM(1), DUM(1), -1,
+     $                   IERR )
+            LWORK_DORGBR_Q_NN = INT( DUM(1) )
+*
+            CALL DORGQR( M, M, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
+            LWORK_DORGQR_MM = INT( DUM(1) )
+*
+            CALL DORGQR( M, N, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
+            LWORK_DORGQR_MN = INT( DUM(1) )
+*
+            CALL DORMBR( 'P', 'R', 'T', N, N, N, DUM(1), N,
+     $                   DUM(1), DUM(1), N, DUM(1), -1, IERR )
+            LWORK_DORMBR_PRT_NN = INT( DUM(1) )
+*
+            CALL DORMBR( 'Q', 'L', 'N', N, N, N, DUM(1), N,
+     $                   DUM(1), DUM(1), N, DUM(1), -1, IERR )
+            LWORK_DORMBR_QLN_NN = INT( DUM(1) )
+*
+            CALL DORMBR( 'Q', 'L', 'N', M, N, N, DUM(1), M,
+     $                   DUM(1), DUM(1), M, DUM(1), -1, IERR )
+            LWORK_DORMBR_QLN_MN = INT( DUM(1) )
+*
+            CALL DORMBR( 'Q', 'L', 'N', M, M, N, DUM(1), M,
+     $                   DUM(1), DUM(1), M, DUM(1), -1, IERR )
+            LWORK_DORMBR_QLN_MM = INT( DUM(1) )
+*
             IF( M.GE.MNTHR ) THEN
                IF( WNTQN ) THEN
 *
-*                 Path 1 (M much larger than N, JOBZ='N')
+*                 Path 1 (M >> N, JOBZ='N')
 *
-                  WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1,
-     $                    -1 )
-                  WRKBL = MAX( WRKBL, 3*N+2*N*
-     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
-                  MAXWRK = MAX( WRKBL, BDSPAC+N )
+                  WRKBL = N + LWORK_DGEQRF_MN
+                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
+                  MAXWRK = MAX( WRKBL, BDSPAC + N )
                   MINWRK = BDSPAC + N
                ELSE IF( WNTQO ) THEN
 *
-*                 Path 2 (M much larger than N, JOBZ='O')
-*
-                  WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
-                  WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
-     $                    N, N, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*N+2*N*
-     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*N+N*
-     $                    ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*N+N*
-     $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
-                  WRKBL = MAX( WRKBL, BDSPAC+3*N )
+*                 Path 2 (M >> N, JOBZ='O')
+*
+                  WRKBL = N + LWORK_DGEQRF_MN
+                  WRKBL = MAX( WRKBL,   N + LWORK_DORGQR_MN )
+                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
+                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
+                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
+                  WRKBL = MAX( WRKBL, 3*N + BDSPAC )
                   MAXWRK = WRKBL + 2*N*N
                   MINWRK = BDSPAC + 2*N*N + 3*N
                ELSE IF( WNTQS ) THEN
 *
-*                 Path 3 (M much larger than N, JOBZ='S')
-*
-                  WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
-                  WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
-     $                    N, N, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*N+2*N*
-     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*N+N*
-     $                    ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*N+N*
-     $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
-                  WRKBL = MAX( WRKBL, BDSPAC+3*N )
+*                 Path 3 (M >> N, JOBZ='S')
+*
+                  WRKBL = N + LWORK_DGEQRF_MN
+                  WRKBL = MAX( WRKBL,   N + LWORK_DORGQR_MN )
+                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
+                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
+                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
+                  WRKBL = MAX( WRKBL, 3*N + BDSPAC )
                   MAXWRK = WRKBL + N*N
                   MINWRK = BDSPAC + N*N + 3*N
                ELSE IF( WNTQA ) THEN
 *
-*                 Path 4 (M much larger than N, JOBZ='A')
-*
-                  WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
-                  WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'DORGQR', ' ', M,
-     $                    M, N, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*N+2*N*
-     $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*N+N*
-     $                    ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*N+N*
-     $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
-                  WRKBL = MAX( WRKBL, BDSPAC+3*N )
+*                 Path 4 (M >> N, JOBZ='A')
+*
+                  WRKBL = N + LWORK_DGEQRF_MN
+                  WRKBL = MAX( WRKBL,   N + LWORK_DORGQR_MM )
+                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
+                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
+                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
+                  WRKBL = MAX( WRKBL, 3*N + BDSPAC )
                   MAXWRK = WRKBL + N*N
-                  MINWRK = BDSPAC + N*N + 2*N + M
+                  MINWRK = N*N + MAX( 3*N + BDSPAC, N + M )
                END IF
             ELSE
 *
-*              Path 5 (M at least N, but not much larger)
+*              Path 5 (M >= N, but not much larger)
 *
-               WRKBL = 3*N + ( M+N )*ILAENV( 1, 'DGEBRD', ' ', M, N, -1,
-     $                 -1 )
+               WRKBL = 3*N + LWORK_DGEBRD_MN
                IF( WNTQN ) THEN
-                  MAXWRK = MAX( WRKBL, BDSPAC+3*N )
+*                 Path 5n (M >= N, jobz='N')
+                  MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
                   MINWRK = 3*N + MAX( M, BDSPAC )
                ELSE IF( WNTQO ) THEN
-                  WRKBL = MAX( WRKBL, 3*N+N*
-     $                    ILAENV( 1, 'DORMBR', 'QLN', M, N, N, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*N+N*
-     $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
-                  WRKBL = MAX( WRKBL, BDSPAC+3*N )
+*                 Path 5o (M >= N, jobz='O')
+                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
+                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MN )
+                  WRKBL = MAX( WRKBL, 3*N + BDSPAC )
                   MAXWRK = WRKBL + M*N
-                  MINWRK = 3*N + MAX( M, N*N+BDSPAC )
+                  MINWRK = 3*N + MAX( M, N*N + BDSPAC )
                ELSE IF( WNTQS ) THEN
-                  WRKBL = MAX( WRKBL, 3*N+N*
-     $                    ILAENV( 1, 'DORMBR', 'QLN', M, N, N, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*N+N*
-     $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
-                  MAXWRK = MAX( WRKBL, BDSPAC+3*N )
+*                 Path 5s (M >= N, jobz='S')
+                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MN )
+                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
+                  MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
                   MINWRK = 3*N + MAX( M, BDSPAC )
                ELSE IF( WNTQA ) THEN
-                  WRKBL = MAX( WRKBL, 3*N+M*
-     $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*N+N*
-     $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
-                  MAXWRK = MAX( MAXWRK, BDSPAC+3*N )
+*                 Path 5a (M >= N, jobz='A')
+                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MM )
+                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
+                  MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
                   MINWRK = 3*N + MAX( M, BDSPAC )
                END IF
             END IF
@@ -410,106 +447,129 @@
 *
 *           Compute space needed for DBDSDC
 *
-            MNTHR = INT( MINMN*11.0D0 / 6.0D0 )
             IF( WNTQN ) THEN
+*              dbdsdc needs only 4*N (or 6*N for uplo=L for LAPACK <= 3.6)
+*              keep 7*N for backwards compatability.
                BDSPAC = 7*M
             ELSE
                BDSPAC = 3*M*M + 4*M
             END IF
+*
+*           Compute space preferred for each routine
+            CALL DGEBRD( M, N, DUM(1), M, DUM(1), DUM(1), DUM(1),
+     $                   DUM(1), DUM(1), -1, IERR )
+            LWORK_DGEBRD_MN = INT( DUM(1) )
+*
+            CALL DGEBRD( M, M, A, M, S, DUM(1), DUM(1),
+     $                   DUM(1), DUM(1), -1, IERR )
+            LWORK_DGEBRD_MM = INT( DUM(1) )
+*
+            CALL DGELQF( M, N, A, M, DUM(1), DUM(1), -1, IERR )
+            LWORK_DGELQF_MN = INT( DUM(1) )
+*
+            CALL DORGLQ( N, N, M, DUM(1), N, DUM(1), DUM(1), -1, IERR )
+            LWORK_DORGLQ_NN = INT( DUM(1) )
+*
+            CALL DORGLQ( M, N, M, A, M, DUM(1), DUM(1), -1, IERR )
+            LWORK_DORGLQ_MN = INT( DUM(1) )
+*
+            CALL DORGBR( 'P', M, M, M, A, N, DUM(1), DUM(1), -1, IERR )
+            LWORK_DORGBR_P_MM = INT( DUM(1) )
+*
+            CALL DORMBR( 'P', 'R', 'T', M, M, M, DUM(1), M,
+     $                   DUM(1), DUM(1), M, DUM(1), -1, IERR )
+            LWORK_DORMBR_PRT_MM = INT( DUM(1) )
+*
+            CALL DORMBR( 'P', 'R', 'T', M, N, M, DUM(1), M,
+     $                   DUM(1), DUM(1), M, DUM(1), -1, IERR )
+            LWORK_DORMBR_PRT_MN = INT( DUM(1) )
+*
+            CALL DORMBR( 'P', 'R', 'T', N, N, M, DUM(1), N,
+     $                   DUM(1), DUM(1), N, DUM(1), -1, IERR )
+            LWORK_DORMBR_PRT_NN = INT( DUM(1) )
+*
+            CALL DORMBR( 'Q', 'L', 'N', M, M, M, DUM(1), M,
+     $                   DUM(1), DUM(1), M, DUM(1), -1, IERR )
+            LWORK_DORMBR_QLN_MM = INT( DUM(1) )
+*
             IF( N.GE.MNTHR ) THEN
                IF( WNTQN ) THEN
 *
-*                 Path 1t (N much larger than M, JOBZ='N')
+*                 Path 1t (N >> M, JOBZ='N')
 *
-                  WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1,
-     $                    -1 )
-                  WRKBL = MAX( WRKBL, 3*M+2*M*
-     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
-                  MAXWRK = MAX( WRKBL, BDSPAC+M )
+                  WRKBL = M + LWORK_DGELQF_MN
+                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
+                  MAXWRK = MAX( WRKBL, BDSPAC + M )
                   MINWRK = BDSPAC + M
                ELSE IF( WNTQO ) THEN
 *
-*                 Path 2t (N much larger than M, JOBZ='O')
-*
-                  WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
-                  WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
-     $                    N, M, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*M+2*M*
-     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*M+M*
-     $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, M, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*M+M*
-     $                    ILAENV( 1, 'DORMBR', 'PRT', M, M, M, -1 ) )
-                  WRKBL = MAX( WRKBL, BDSPAC+3*M )
+*                 Path 2t (N >> M, JOBZ='O')
+*
+                  WRKBL = M + LWORK_DGELQF_MN
+                  WRKBL = MAX( WRKBL,   M + LWORK_DORGLQ_MN )
+                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
+                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
+                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
+                  WRKBL = MAX( WRKBL, 3*M + BDSPAC )
                   MAXWRK = WRKBL + 2*M*M
                   MINWRK = BDSPAC + 2*M*M + 3*M
                ELSE IF( WNTQS ) THEN
 *
-*                 Path 3t (N much larger than M, JOBZ='S')
-*
-                  WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
-                  WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
-     $                    N, M, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*M+2*M*
-     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*M+M*
-     $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, M, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*M+M*
-     $                    ILAENV( 1, 'DORMBR', 'PRT', M, M, M, -1 ) )
-                  WRKBL = MAX( WRKBL, BDSPAC+3*M )
+*                 Path 3t (N >> M, JOBZ='S')
+*
+                  WRKBL = M + LWORK_DGELQF_MN
+                  WRKBL = MAX( WRKBL,   M + LWORK_DORGLQ_MN )
+                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
+                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
+                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
+                  WRKBL = MAX( WRKBL, 3*M + BDSPAC )
                   MAXWRK = WRKBL + M*M
                   MINWRK = BDSPAC + M*M + 3*M
                ELSE IF( WNTQA ) THEN
 *
-*                 Path 4t (N much larger than M, JOBZ='A')
-*
-                  WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
-                  WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'DORGLQ', ' ', N,
-     $                    N, M, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*M+2*M*
-     $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*M+M*
-     $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, M, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*M+M*
-     $                    ILAENV( 1, 'DORMBR', 'PRT', M, M, M, -1 ) )
-                  WRKBL = MAX( WRKBL, BDSPAC+3*M )
+*                 Path 4t (N >> M, JOBZ='A')
+*
+                  WRKBL = M + LWORK_DGELQF_MN
+                  WRKBL = MAX( WRKBL,   M + LWORK_DORGLQ_NN )
+                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
+                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
+                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
+                  WRKBL = MAX( WRKBL, 3*M + BDSPAC )
                   MAXWRK = WRKBL + M*M
-                  MINWRK = BDSPAC + M*M + 3*M
+                  MINWRK = M*M + MAX( 3*M + BDSPAC, M + N )
                END IF
             ELSE
 *
-*              Path 5t (N greater than M, but not much larger)
+*              Path 5t (N > M, but not much larger)
 *
-               WRKBL = 3*M + ( M+N )*ILAENV( 1, 'DGEBRD', ' ', M, N, -1,
-     $                 -1 )
+               WRKBL = 3*M + LWORK_DGEBRD_MN
                IF( WNTQN ) THEN
-                  MAXWRK = MAX( WRKBL, BDSPAC+3*M )
+*                 Path 5tn (N > M, jobz='N')
+                  MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
                   MINWRK = 3*M + MAX( N, BDSPAC )
                ELSE IF( WNTQO ) THEN
-                  WRKBL = MAX( WRKBL, 3*M+M*
-     $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*M+M*
-     $                    ILAENV( 1, 'DORMBR', 'PRT', M, N, M, -1 ) )
-                  WRKBL = MAX( WRKBL, BDSPAC+3*M )
+*                 Path 5to (N > M, jobz='O')
+                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
+                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MN )
+                  WRKBL = MAX( WRKBL, 3*M + BDSPAC )
                   MAXWRK = WRKBL + M*N
-                  MINWRK = 3*M + MAX( N, M*M+BDSPAC )
+                  MINWRK = 3*M + MAX( N, M*M + BDSPAC )
                ELSE IF( WNTQS ) THEN
-                  WRKBL = MAX( WRKBL, 3*M+M*
-     $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*M+M*
-     $                    ILAENV( 1, 'DORMBR', 'PRT', M, N, M, -1 ) )
-                  MAXWRK = MAX( WRKBL, BDSPAC+3*M )
+*                 Path 5ts (N > M, jobz='S')
+                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
+                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MN )
+                  MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
                   MINWRK = 3*M + MAX( N, BDSPAC )
                ELSE IF( WNTQA ) THEN
-                  WRKBL = MAX( WRKBL, 3*M+M*
-     $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) )
-                  WRKBL = MAX( WRKBL, 3*M+M*
-     $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, M, -1 ) )
-                  MAXWRK = MAX( WRKBL, BDSPAC+3*M )
+*                 Path 5ta (N > M, jobz='A')
+                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
+                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_NN )
+                  MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
                   MINWRK = 3*M + MAX( N, BDSPAC )
                END IF
             END IF
          END IF
+         
          MAXWRK = MAX( MAXWRK, MINWRK )
          WORK( 1 ) = MAXWRK
 *
@@ -559,17 +619,18 @@
 *
             IF( WNTQN ) THEN
 *
-*              Path 1 (M much larger than N, JOBZ='N')
+*              Path 1 (M >> N, JOBZ='N')
 *              No singular vectors to be computed
 *
                ITAU = 1
                NWORK = ITAU + N
 *
 *              Compute A=Q*R
-*              (Workspace: need 2*N, prefer N+N*NB)
+*              Workspace: need   N [tau] + N    [work]
+*              Workspace: prefer N [tau] + N*NB [work]
 *
                CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
 *              Zero out below R
 *
@@ -580,7 +641,8 @@
                NWORK = ITAUP + N
 *
 *              Bidiagonalize R in A
-*              (Workspace: need 4*N, prefer 3*N+2*N*NB)
+*              Workspace: need   3*N [e, tauq, taup] + N      [work]
+*              Workspace: prefer 3*N [e, tauq, taup] + 2*N*NB [work]
 *
                CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
      $                      WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
@@ -588,14 +650,14 @@
                NWORK = IE + N
 *
 *              Perform bidiagonal SVD, computing singular values only
-*              (Workspace: need N+BDSPAC)
+*              Workspace: need   N [e] + BDSPAC
 *
                CALL DBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM, 1,
      $                      DUM, IDUM, WORK( NWORK ), IWORK, INFO )
 *
             ELSE IF( WNTQO ) THEN
 *
-*              Path 2 (M much larger than N, JOBZ = 'O')
+*              Path 2 (M >> N, JOBZ = 'O')
 *              N left singular vectors to be overwritten on A and
 *              N right singular vectors to be computed in VT
 *
@@ -603,42 +665,45 @@
 *
 *              WORK(IR) is LDWRKR by N
 *
-               IF( LWORK.GE.LDA*N+N*N+3*N+BDSPAC ) THEN
+               IF( LWORK .GE. LDA*N + N*N + 3*N + BDSPAC ) THEN
                   LDWRKR = LDA
                ELSE
-                  LDWRKR = ( LWORK-N*N-3*N-BDSPAC ) / N
+                  LDWRKR = ( LWORK - N*N - 3*N - BDSPAC ) / N
                END IF
                ITAU = IR + LDWRKR*N
                NWORK = ITAU + N
 *
 *              Compute A=Q*R
-*              (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+*              Workspace: need   N*N [R] + N [tau] + N    [work]
+*              Workspace: prefer N*N [R] + N [tau] + N*NB [work]
 *
                CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
 *              Copy R to WORK(IR), zeroing out below it
 *
                CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
-               CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, WORK( IR+1 ),
+               CALL DLASET( 'L', N - 1, N - 1, ZERO, ZERO, WORK(IR+1),
      $                      LDWRKR )
 *
 *              Generate Q in A
-*              (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+*              Workspace: need   N*N [R] + N [tau] + N    [work]
+*              Workspace: prefer N*N [R] + N [tau] + N*NB [work]
 *
                CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
-     $                      WORK( NWORK ), LWORK-NWORK+1, IERR )
+     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
                IE = ITAU
                ITAUQ = IE + N
                ITAUP = ITAUQ + N
                NWORK = ITAUP + N
 *
-*              Bidiagonalize R in VT, copying result to WORK(IR)
-*              (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
+*              Bidiagonalize R in WORK(IR)
+*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N      [work]
+*              Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + 2*N*NB [work]
 *
                CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
      $                      WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
 *              WORK(IU) is N by N
 *
@@ -648,7 +713,7 @@
 *              Perform bidiagonal SVD, computing left singular vectors
 *              of bidiagonal matrix in WORK(IU) and computing right
 *              singular vectors of bidiagonal matrix in VT
-*              (Workspace: need N+N*N+BDSPAC)
+*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N*N [U] + BDSPAC
 *
                CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ), N,
      $                      VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
@@ -656,21 +721,23 @@
 *
 *              Overwrite WORK(IU) by left singular vectors of R
 *              and VT by right singular vectors of R
-*              (Workspace: need 2*N*N+3*N, prefer 2*N*N+2*N+N*NB)
+*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N*N [U] + N    [work]
+*              Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
 *
                CALL DORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR,
      $                      WORK( ITAUQ ), WORK( IU ), N, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
                CALL DORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ), LDWRKR,
      $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
 *              Multiply Q in A by left singular vectors of R in
 *              WORK(IU), storing result in WORK(IR) and copying to A
-*              (Workspace: need 2*N*N, prefer N*N+M*N)
+*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N*N [U]
+*              Workspace: prefer M*N [R] + 3*N [e, tauq, taup] + N*N [U]
 *
                DO 10 I = 1, M, LDWRKR
-                  CHUNK = MIN( M-I+1, LDWRKR )
+                  CHUNK = MIN( M - I + 1, LDWRKR )
                   CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ),
      $                        LDA, WORK( IU ), N, ZERO, WORK( IR ),
      $                        LDWRKR )
@@ -680,7 +747,7 @@
 *
             ELSE IF( WNTQS ) THEN
 *
-*              Path 3 (M much larger than N, JOBZ='S')
+*              Path 3 (M >> N, JOBZ='S')
 *              N left singular vectors to be computed in U and
 *              N right singular vectors to be computed in VT
 *
@@ -693,38 +760,41 @@
                NWORK = ITAU + N
 *
 *              Compute A=Q*R
-*              (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+*              Workspace: need   N*N [R] + N [tau] + N    [work]
+*              Workspace: prefer N*N [R] + N [tau] + N*NB [work]
 *
                CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
 *              Copy R to WORK(IR), zeroing out below it
 *
                CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
-               CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, WORK( IR+1 ),
+               CALL DLASET( 'L', N - 1, N - 1, ZERO, ZERO, WORK(IR+1),
      $                      LDWRKR )
 *
 *              Generate Q in A
-*              (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+*              Workspace: need   N*N [R] + N [tau] + N    [work]
+*              Workspace: prefer N*N [R] + N [tau] + N*NB [work]
 *
                CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
-     $                      WORK( NWORK ), LWORK-NWORK+1, IERR )
+     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
                IE = ITAU
                ITAUQ = IE + N
                ITAUP = ITAUQ + N
                NWORK = ITAUP + N
 *
 *              Bidiagonalize R in WORK(IR)
-*              (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
+*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N      [work]
+*              Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + 2*N*NB [work]
 *
                CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
      $                      WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
 *              Perform bidiagonal SVD, computing left singular vectors
 *              of bidiagoal matrix in U and computing right singular
 *              vectors of bidiagonal matrix in VT
-*              (Workspace: need N+BDSPAC)
+*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + BDSPAC
 *
                CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
      $                      LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
@@ -732,19 +802,20 @@
 *
 *              Overwrite U by left singular vectors of R and VT
 *              by right singular vectors of R
-*              (Workspace: need N*N+3*N, prefer N*N+2*N+N*NB)
+*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N    [work]
+*              Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + N*NB [work]
 *
                CALL DORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR,
      $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
                CALL DORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ), LDWRKR,
      $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
 *              Multiply Q in A by left singular vectors of R in
 *              WORK(IR), storing result in U
-*              (Workspace: need N*N)
+*              Workspace: need   N*N [R]
 *
                CALL DLACPY( 'F', N, N, U, LDU, WORK( IR ), LDWRKR )
                CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA, WORK( IR ),
@@ -752,7 +823,7 @@
 *
             ELSE IF( WNTQA ) THEN
 *
-*              Path 4 (M much larger than N, JOBZ='A')
+*              Path 4 (M >> N, JOBZ='A')
 *              M left singular vectors to be computed in U and
 *              N right singular vectors to be computed in VT
 *
@@ -765,16 +836,18 @@
                NWORK = ITAU + N
 *
 *              Compute A=Q*R, copying result to U
-*              (Workspace: need N*N+N+M, prefer N*N+N+M*NB)
+*              Workspace: need   N*N [U] + N [tau] + N    [work]
+*              Workspace: prefer N*N [U] + N [tau] + N*NB [work]
 *
                CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
                CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
 *
 *              Generate Q in U
-*              (Workspace: need N*N+N+M, prefer N*N+N+M*NB)
+*              Workspace: need   N*N [U] + N [tau] + M    [work]
+*              Workspace: prefer N*N [U] + N [tau] + M*NB [work]
                CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
-     $                      WORK( NWORK ), LWORK-NWORK+1, IERR )
+     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
 *
 *              Produce R in A, zeroing out other entries
 *
@@ -785,7 +858,8 @@
                NWORK = ITAUP + N
 *
 *              Bidiagonalize R in A
-*              (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
+*              Workspace: need   N*N [U] + 3*N [e, tauq, taup] + N      [work]
+*              Workspace: prefer N*N [U] + 3*N [e, tauq, taup] + 2*N*NB [work]
 *
                CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
      $                      WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
@@ -794,7 +868,7 @@
 *              Perform bidiagonal SVD, computing left singular vectors
 *              of bidiagonal matrix in WORK(IU) and computing right
 *              singular vectors of bidiagonal matrix in VT
-*              (Workspace: need N+N*N+BDSPAC)
+*              Workspace: need   N*N [U] + 3*N [e, tauq, taup] + BDSPAC
 *
                CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ), N,
      $                      VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
@@ -802,18 +876,19 @@
 *
 *              Overwrite WORK(IU) by left singular vectors of R and VT
 *              by right singular vectors of R
-*              (Workspace: need N*N+3*N, prefer N*N+2*N+N*NB)
+*              Workspace: need   N*N [U] + 3*N [e, tauq, taup] + N    [work]
+*              Workspace: prefer N*N [U] + 3*N [e, tauq, taup] + N*NB [work]
 *
                CALL DORMBR( 'Q', 'L', 'N', N, N, N, A, LDA,
      $                      WORK( ITAUQ ), WORK( IU ), LDWRKU,
-     $                      WORK( NWORK ), LWORK-NWORK+1, IERR )
+     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
                CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
      $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
 *              Multiply Q in U by left singular vectors of R in
 *              WORK(IU), storing result in A
-*              (Workspace: need N*N)
+*              Workspace: need   N*N [U]
 *
                CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU, WORK( IU ),
      $                     LDWRKU, ZERO, A, LDA )
@@ -828,7 +903,7 @@
 *
 *           M .LT. MNTHR
 *
-*           Path 5 (M at least N, but not much larger)
+*           Path 5 (M >= N, but not much larger)
 *           Reduce to bidiagonal form without QR decomposition
 *
             IE = 1
@@ -837,21 +912,24 @@
             NWORK = ITAUP + N
 *
 *           Bidiagonalize A
-*           (Workspace: need 3*N+M, prefer 3*N+(M+N)*NB)
+*           Workspace: need   3*N [e, tauq, taup] + M        [work]
+*           Workspace: prefer 3*N [e, tauq, taup] + (M+N)*NB [work]
 *
             CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
      $                   WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
      $                   IERR )
             IF( WNTQN ) THEN
 *
+*              Path 5n (M >= N, JOBZ='N')
 *              Perform bidiagonal SVD, only computing singular values
-*              (Workspace: need N+BDSPAC)
+*              Workspace: need   3*N [e, tauq, taup] + BDSPAC
 *
                CALL DBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM, 1,
      $                      DUM, IDUM, WORK( NWORK ), IWORK, INFO )
             ELSE IF( WNTQO ) THEN
+*              Path 5o (M >= N, JOBZ='O')
                IU = NWORK
-               IF( LWORK.GE.M*N+3*N+BDSPAC ) THEN
+               IF( LWORK .GE. M*N + 3*N + BDSPAC ) THEN
 *
 *                 WORK( IU ) is M by N
 *
@@ -859,6 +937,8 @@
                   NWORK = IU + LDWRKU*N
                   CALL DLASET( 'F', M, N, ZERO, ZERO, WORK( IU ),
      $                         LDWRKU )
+*                 IR is unused; silence compile warnings
+                  IR = -1
                ELSE
 *
 *                 WORK( IU ) is N by N
@@ -869,53 +949,59 @@
 *                 WORK(IR) is LDWRKR by N
 *
                   IR = NWORK
-                  LDWRKR = ( LWORK-N*N-3*N ) / N
+                  LDWRKR = ( LWORK - N*N - 3*N ) / N
                END IF
                NWORK = IU + LDWRKU*N
 *
 *              Perform bidiagonal SVD, computing left singular vectors
 *              of bidiagonal matrix in WORK(IU) and computing right
 *              singular vectors of bidiagonal matrix in VT
-*              (Workspace: need N+N*N+BDSPAC)
+*              Workspace: need   3*N [e, tauq, taup] + N*N [U] + BDSPAC
 *
                CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ),
      $                      LDWRKU, VT, LDVT, DUM, IDUM, WORK( NWORK ),
      $                      IWORK, INFO )
 *
 *              Overwrite VT by right singular vectors of A
-*              (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+*              Workspace: need   3*N [e, tauq, taup] + N*N [U] + N    [work]
+*              Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
 *
                CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
      $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
-               IF( LWORK.GE.M*N+3*N+BDSPAC ) THEN
+               IF( LWORK .GE. M*N + 3*N + BDSPAC ) THEN
 *
+*                 Path 5o-fast
 *                 Overwrite WORK(IU) by left singular vectors of A
-*                 (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+*                 Workspace: need   3*N [e, tauq, taup] + M*N [U] + N    [work]
+*                 Workspace: prefer 3*N [e, tauq, taup] + M*N [U] + N*NB [work]
 *
                   CALL DORMBR( 'Q', 'L', 'N', M, N, N, A, LDA,
      $                         WORK( ITAUQ ), WORK( IU ), LDWRKU,
-     $                         WORK( NWORK ), LWORK-NWORK+1, IERR )
+     $                         WORK( NWORK ), LWORK - NWORK + 1, IERR )
 *
 *                 Copy left singular vectors of A from WORK(IU) to A
 *
                   CALL DLACPY( 'F', M, N, WORK( IU ), LDWRKU, A, LDA )
                ELSE
 *
+*                 Path 5o-slow
 *                 Generate Q in A
-*                 (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+*                 Workspace: need   3*N [e, tauq, taup] + N*N [U] + N    [work]
+*                 Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
 *
                   CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
-     $                         WORK( NWORK ), LWORK-NWORK+1, IERR )
+     $                         WORK( NWORK ), LWORK - NWORK + 1, IERR )
 *
 *                 Multiply Q in A by left singular vectors of
 *                 bidiagonal matrix in WORK(IU), storing result in
 *                 WORK(IR) and copying to A
-*                 (Workspace: need 2*N*N, prefer N*N+M*N)
+*                 Workspace: need   3*N [e, tauq, taup] + N*N [U] + NB*N [R]
+*                 Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + M*N  [R]
 *
                   DO 20 I = 1, M, LDWRKR
-                     CHUNK = MIN( M-I+1, LDWRKR )
+                     CHUNK = MIN( M - I + 1, LDWRKR )
                      CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ),
      $                           LDA, WORK( IU ), LDWRKU, ZERO,
      $                           WORK( IR ), LDWRKR )
@@ -926,10 +1012,11 @@
 *
             ELSE IF( WNTQS ) THEN
 *
+*              Path 5s (M >= N, JOBZ='S')
 *              Perform bidiagonal SVD, computing left singular vectors
 *              of bidiagonal matrix in U and computing right singular
 *              vectors of bidiagonal matrix in VT
-*              (Workspace: need N+BDSPAC)
+*              Workspace: need   3*N [e, tauq, taup] + BDSPAC
 *
                CALL DLASET( 'F', M, N, ZERO, ZERO, U, LDU )
                CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
@@ -938,20 +1025,22 @@
 *
 *              Overwrite U by left singular vectors of A and VT
 *              by right singular vectors of A
-*              (Workspace: need 3*N, prefer 2*N+N*NB)
+*              Workspace: need   3*N [e, tauq, taup] + N    [work]
+*              Workspace: prefer 3*N [e, tauq, taup] + N*NB [work]
 *
                CALL DORMBR( 'Q', 'L', 'N', M, N, N, A, LDA,
      $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
                CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
      $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
             ELSE IF( WNTQA ) THEN
 *
+*              Path 5a (M >= N, JOBZ='A')
 *              Perform bidiagonal SVD, computing left singular vectors
 *              of bidiagonal matrix in U and computing right singular
 *              vectors of bidiagonal matrix in VT
-*              (Workspace: need N+BDSPAC)
+*              Workspace: need   3*N [e, tauq, taup] + BDSPAC
 *
                CALL DLASET( 'F', M, M, ZERO, ZERO, U, LDU )
                CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
@@ -961,20 +1050,21 @@
 *              Set the right corner of U to identity matrix
 *
                IF( M.GT.N ) THEN
-                  CALL DLASET( 'F', M-N, M-N, ZERO, ONE, U( N+1, N+1 ),
+                  CALL DLASET( 'F', M - N, M - N, ZERO, ONE, U(N+1,N+1),
      $                         LDU )
                END IF
 *
 *              Overwrite U by left singular vectors of A and VT
 *              by right singular vectors of A
-*              (Workspace: need N*N+2*N+M, prefer N*N+2*N+M*NB)
+*              Workspace: need   3*N [e, tauq, taup] + M    [work]
+*              Workspace: prefer 3*N [e, tauq, taup] + M*NB [work]
 *
                CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
      $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
                CALL DORMBR( 'P', 'R', 'T', N, N, M, A, LDA,
      $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
             END IF
 *
          END IF
@@ -989,17 +1079,18 @@
 *
             IF( WNTQN ) THEN
 *
-*              Path 1t (N much larger than M, JOBZ='N')
+*              Path 1t (N >> M, JOBZ='N')
 *              No singular vectors to be computed
 *
                ITAU = 1
                NWORK = ITAU + M
 *
 *              Compute A=L*Q
-*              (Workspace: need 2*M, prefer M+M*NB)
+*              Workspace: need   M [tau] + M [work]
+*              Workspace: prefer M [tau] + M*NB [work]
 *
                CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
 *              Zero out above L
 *
@@ -1010,7 +1101,8 @@
                NWORK = ITAUP + M
 *
 *              Bidiagonalize L in A
-*              (Workspace: need 4*M, prefer 3*M+2*M*NB)
+*              Workspace: need   3*M [e, tauq, taup] + M      [work]
+*              Workspace: prefer 3*M [e, tauq, taup] + 2*M*NB [work]
 *
                CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
      $                      WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
@@ -1018,68 +1110,69 @@
                NWORK = IE + M
 *
 *              Perform bidiagonal SVD, computing singular values only
-*              (Workspace: need M+BDSPAC)
+*              Workspace: need   M [e] + BDSPAC
 *
                CALL DBDSDC( 'U', 'N', M, S, WORK( IE ), DUM, 1, DUM, 1,
      $                      DUM, IDUM, WORK( NWORK ), IWORK, INFO )
 *
             ELSE IF( WNTQO ) THEN
 *
-*              Path 2t (N much larger than M, JOBZ='O')
+*              Path 2t (N >> M, JOBZ='O')
 *              M right singular vectors to be overwritten on A and
 *              M left singular vectors to be computed in U
 *
                IVT = 1
 *
-*              IVT is M by M
+*              WORK(IVT) is M by M
+*              WORK(IL)  is M by M; it is later resized to M by chunk for gemm
 *
                IL = IVT + M*M
-               IF( LWORK.GE.M*N+M*M+3*M+BDSPAC ) THEN
-*
-*                 WORK(IL) is M by N
-*
+               IF( LWORK .GE. M*N + M*M + 3*M + BDSPAC ) THEN
                   LDWRKL = M
                   CHUNK = N
                ELSE
                   LDWRKL = M
-                  CHUNK = ( LWORK-M*M ) / M
+                  CHUNK = ( LWORK - M*M ) / M
                END IF
                ITAU = IL + LDWRKL*M
                NWORK = ITAU + M
 *
 *              Compute A=L*Q
-*              (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+*              Workspace: need   M*M [VT] + M*M [L] + M [tau] + M    [work]
+*              Workspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
 *
                CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
 *              Copy L to WORK(IL), zeroing about above it
 *
                CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL )
-               CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
-     $                      WORK( IL+LDWRKL ), LDWRKL )
+               CALL DLASET( 'U', M - 1, M - 1, ZERO, ZERO,
+     $                      WORK( IL + LDWRKL ), LDWRKL )
 *
 *              Generate Q in A
-*              (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+*              Workspace: need   M*M [VT] + M*M [L] + M [tau] + M    [work]
+*              Workspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
 *
                CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
-     $                      WORK( NWORK ), LWORK-NWORK+1, IERR )
+     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
                IE = ITAU
                ITAUQ = IE + M
                ITAUP = ITAUQ + M
                NWORK = ITAUP + M
 *
 *              Bidiagonalize L in WORK(IL)
-*              (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
+*              Workspace: need   M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M      [work]
+*              Workspace: prefer M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + 2*M*NB [work]
 *
                CALL DGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ),
      $                      WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
 *              Perform bidiagonal SVD, computing left singular vectors
 *              of bidiagonal matrix in U, and computing right singular
 *              vectors of bidiagonal matrix in WORK(IVT)
-*              (Workspace: need M+M*M+BDSPAC)
+*              Workspace: need   M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + BDSPAC
 *
                CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU,
      $                      WORK( IVT ), M, DUM, IDUM, WORK( NWORK ),
@@ -1087,21 +1180,24 @@
 *
 *              Overwrite U by left singular vectors of L and WORK(IVT)
 *              by right singular vectors of L
-*              (Workspace: need 2*M*M+3*M, prefer 2*M*M+2*M+M*NB)
+*              Workspace: need   M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M    [work]
+*              Workspace: prefer M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M*NB [work]
 *
                CALL DORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL,
      $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
                CALL DORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ), LDWRKL,
      $                      WORK( ITAUP ), WORK( IVT ), M,
-     $                      WORK( NWORK ), LWORK-NWORK+1, IERR )
+     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
 *
 *              Multiply right singular vectors of L in WORK(IVT) by Q
 *              in A, storing result in WORK(IL) and copying to A
-*              (Workspace: need 2*M*M, prefer M*M+M*N)
+*              Workspace: need   M*M [VT] + M*M [L]
+*              Workspace: prefer M*M [VT] + M*N [L]
+*              At this point, L is resized as M by chunk.
 *
                DO 30 I = 1, N, CHUNK
-                  BLK = MIN( N-I+1, CHUNK )
+                  BLK = MIN( N - I + 1, CHUNK )
                   CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IVT ), M,
      $                        A( 1, I ), LDA, ZERO, WORK( IL ), LDWRKL )
                   CALL DLACPY( 'F', M, BLK, WORK( IL ), LDWRKL,
@@ -1110,7 +1206,7 @@
 *
             ELSE IF( WNTQS ) THEN
 *
-*              Path 3t (N much larger than M, JOBZ='S')
+*              Path 3t (N >> M, JOBZ='S')
 *              M right singular vectors to be computed in VT and
 *              M left singular vectors to be computed in U
 *
@@ -1123,38 +1219,41 @@
                NWORK = ITAU + M
 *
 *              Compute A=L*Q
-*              (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+*              Workspace: need   M*M [L] + M [tau] + M    [work]
+*              Workspace: prefer M*M [L] + M [tau] + M*NB [work]
 *
                CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
 *              Copy L to WORK(IL), zeroing out above it
 *
                CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL )
-               CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
-     $                      WORK( IL+LDWRKL ), LDWRKL )
+               CALL DLASET( 'U', M - 1, M - 1, ZERO, ZERO,
+     $                      WORK( IL + LDWRKL ), LDWRKL )
 *
 *              Generate Q in A
-*              (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+*              Workspace: need   M*M [L] + M [tau] + M    [work]
+*              Workspace: prefer M*M [L] + M [tau] + M*NB [work]
 *
                CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
-     $                      WORK( NWORK ), LWORK-NWORK+1, IERR )
+     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
                IE = ITAU
                ITAUQ = IE + M
                ITAUP = ITAUQ + M
                NWORK = ITAUP + M
 *
-*              Bidiagonalize L in WORK(IU), copying result to U
-*              (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
+*              Bidiagonalize L in WORK(IU).
+*              Workspace: need   M*M [L] + 3*M [e, tauq, taup] + M      [work]
+*              Workspace: prefer M*M [L] + 3*M [e, tauq, taup] + 2*M*NB [work]
 *
                CALL DGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ),
      $                      WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
 *              Perform bidiagonal SVD, computing left singular vectors
 *              of bidiagonal matrix in U and computing right singular
 *              vectors of bidiagonal matrix in VT
-*              (Workspace: need M+BDSPAC)
+*              Workspace: need   M*M [L] + 3*M [e, tauq, taup] + BDSPAC
 *
                CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU, VT,
      $                      LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
@@ -1162,18 +1261,19 @@
 *
 *              Overwrite U by left singular vectors of L and VT
 *              by right singular vectors of L
-*              (Workspace: need M*M+3*M, prefer M*M+2*M+M*NB)
+*              Workspace: need   M*M [L] + 3*M [e, tauq, taup] + M    [work]
+*              Workspace: prefer M*M [L] + 3*M [e, tauq, taup] + M*NB [work]
 *
                CALL DORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL,
      $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
                CALL DORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ), LDWRKL,
      $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
 *              Multiply right singular vectors of L in WORK(IL) by
 *              Q in A, storing result in VT
-*              (Workspace: need M*M)
+*              Workspace: need   M*M [L]
 *
                CALL DLACPY( 'F', M, M, VT, LDVT, WORK( IL ), LDWRKL )
                CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IL ), LDWRKL,
@@ -1181,7 +1281,7 @@
 *
             ELSE IF( WNTQA ) THEN
 *
-*              Path 4t (N much larger than M, JOBZ='A')
+*              Path 4t (N >> M, JOBZ='A')
 *              N right singular vectors to be computed in VT and
 *              M left singular vectors to be computed in U
 *
@@ -1194,17 +1294,19 @@
                NWORK = ITAU + M
 *
 *              Compute A=L*Q, copying result to VT
-*              (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+*              Workspace: need   M*M [VT] + M [tau] + M    [work]
+*              Workspace: prefer M*M [VT] + M [tau] + M*NB [work]
 *
                CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
                CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
 *
 *              Generate Q in VT
-*              (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+*              Workspace: need   M*M [VT] + M [tau] + N    [work]
+*              Workspace: prefer M*M [VT] + M [tau] + N*NB [work]
 *
                CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
-     $                      WORK( NWORK ), LWORK-NWORK+1, IERR )
+     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
 *
 *              Produce L in A, zeroing out other entries
 *
@@ -1215,7 +1317,8 @@
                NWORK = ITAUP + M
 *
 *              Bidiagonalize L in A
-*              (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
+*              Workspace: need   M*M [VT] + 3*M [e, tauq, taup] + M      [work]
+*              Workspace: prefer M*M [VT] + 3*M [e, tauq, taup] + 2*M*NB [work]
 *
                CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
      $                      WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
@@ -1224,7 +1327,7 @@
 *              Perform bidiagonal SVD, computing left singular vectors
 *              of bidiagonal matrix in U and computing right singular
 *              vectors of bidiagonal matrix in WORK(IVT)
-*              (Workspace: need M+M*M+BDSPAC)
+*              Workspace: need   M*M [VT] + 3*M [e, tauq, taup] + BDSPAC
 *
                CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU,
      $                      WORK( IVT ), LDWKVT, DUM, IDUM,
@@ -1232,18 +1335,19 @@
 *
 *              Overwrite U by left singular vectors of L and WORK(IVT)
 *              by right singular vectors of L
-*              (Workspace: need M*M+3*M, prefer M*M+2*M+M*NB)
+*              Workspace: need   M*M [VT] + 3*M [e, tauq, taup]+ M    [work]
+*              Workspace: prefer M*M [VT] + 3*M [e, tauq, taup]+ M*NB [work]
 *
                CALL DORMBR( 'Q', 'L', 'N', M, M, M, A, LDA,
      $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
                CALL DORMBR( 'P', 'R', 'T', M, M, M, A, LDA,
      $                      WORK( ITAUP ), WORK( IVT ), LDWKVT,
-     $                      WORK( NWORK ), LWORK-NWORK+1, IERR )
+     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
 *
 *              Multiply right singular vectors of L in WORK(IVT) by
 *              Q in VT, storing result in A
-*              (Workspace: need M*M)
+*              Workspace: need   M*M [VT]
 *
                CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IVT ), LDWKVT,
      $                     VT, LDVT, ZERO, A, LDA )
@@ -1258,7 +1362,7 @@
 *
 *           N .LT. MNTHR
 *
-*           Path 5t (N greater than M, but not much larger)
+*           Path 5t (N > M, but not much larger)
 *           Reduce to bidiagonal form without LQ decomposition
 *
             IE = 1
@@ -1267,28 +1371,33 @@
             NWORK = ITAUP + M
 *
 *           Bidiagonalize A
-*           (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
+*           Workspace: need   3*M [e, tauq, taup] + N        [work]
+*           Workspace: prefer 3*M [e, tauq, taup] + (M+N)*NB [work]
 *
             CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
      $                   WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
      $                   IERR )
             IF( WNTQN ) THEN
 *
+*              Path 5tn (N > M, JOBZ='N')
 *              Perform bidiagonal SVD, only computing singular values
-*              (Workspace: need M+BDSPAC)
+*              Workspace: need   3*M [e, tauq, taup] + BDSPAC
 *
                CALL DBDSDC( 'L', 'N', M, S, WORK( IE ), DUM, 1, DUM, 1,
      $                      DUM, IDUM, WORK( NWORK ), IWORK, INFO )
             ELSE IF( WNTQO ) THEN
+*              Path 5to (N > M, JOBZ='O')
                LDWKVT = M
                IVT = NWORK
-               IF( LWORK.GE.M*N+3*M+BDSPAC ) THEN
+               IF( LWORK .GE. M*N + 3*M + BDSPAC ) THEN
 *
 *                 WORK( IVT ) is M by N
 *
                   CALL DLASET( 'F', M, N, ZERO, ZERO, WORK( IVT ),
      $                         LDWKVT )
                   NWORK = IVT + LDWKVT*N
+*                 IL is unused; silence compile warnings
+                  IL = -1
                ELSE
 *
 *                 WORK( IVT ) is M by M
@@ -1298,52 +1407,58 @@
 *
 *                 WORK(IL) is M by CHUNK
 *
-                  CHUNK = ( LWORK-M*M-3*M ) / M
+                  CHUNK = ( LWORK - M*M - 3*M ) / M
                END IF
 *
 *              Perform bidiagonal SVD, computing left singular vectors
 *              of bidiagonal matrix in U and computing right singular
 *              vectors of bidiagonal matrix in WORK(IVT)
-*              (Workspace: need M*M+BDSPAC)
+*              Workspace: need   3*M [e, tauq, taup] + M*M [VT] + BDSPAC
 *
                CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU,
      $                      WORK( IVT ), LDWKVT, DUM, IDUM,
      $                      WORK( NWORK ), IWORK, INFO )
 *
 *              Overwrite U by left singular vectors of A
-*              (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+*              Workspace: need   3*M [e, tauq, taup] + M*M [VT] + M    [work]
+*              Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*NB [work]
 *
                CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
      $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
 *
-               IF( LWORK.GE.M*N+3*M+BDSPAC ) THEN
+               IF( LWORK .GE. M*N + 3*M + BDSPAC ) THEN
 *
+*                 Path 5to-fast
 *                 Overwrite WORK(IVT) by left singular vectors of A
-*                 (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+*                 Workspace: need   3*M [e, tauq, taup] + M*N [VT] + M    [work]
+*                 Workspace: prefer 3*M [e, tauq, taup] + M*N [VT] + M*NB [work]
 *
                   CALL DORMBR( 'P', 'R', 'T', M, N, M, A, LDA,
      $                         WORK( ITAUP ), WORK( IVT ), LDWKVT,
-     $                         WORK( NWORK ), LWORK-NWORK+1, IERR )
+     $                         WORK( NWORK ), LWORK - NWORK + 1, IERR )
 *
 *                 Copy right singular vectors of A from WORK(IVT) to A
 *
                   CALL DLACPY( 'F', M, N, WORK( IVT ), LDWKVT, A, LDA )
                ELSE
 *
+*                 Path 5to-slow
 *                 Generate P**T in A
-*                 (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+*                 Workspace: need   3*M [e, tauq, taup] + M*M [VT] + M    [work]
+*                 Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*NB [work]
 *
                   CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
-     $                         WORK( NWORK ), LWORK-NWORK+1, IERR )
+     $                         WORK( NWORK ), LWORK - NWORK + 1, IERR )
 *
 *                 Multiply Q in A by right singular vectors of
 *                 bidiagonal matrix in WORK(IVT), storing result in
 *                 WORK(IL) and copying to A
-*                 (Workspace: need 2*M*M, prefer M*M+M*N)
+*                 Workspace: need   3*M [e, tauq, taup] + M*M [VT] + M*NB [L]
+*                 Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*N  [L]
 *
                   DO 40 I = 1, N, CHUNK
-                     BLK = MIN( N-I+1, CHUNK )
+                     BLK = MIN( N - I + 1, CHUNK )
                      CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IVT ),
      $                           LDWKVT, A( 1, I ), LDA, ZERO,
      $                           WORK( IL ), M )
@@ -1353,10 +1468,11 @@
                END IF
             ELSE IF( WNTQS ) THEN
 *
+*              Path 5ts (N > M, JOBZ='S')
 *              Perform bidiagonal SVD, computing left singular vectors
 *              of bidiagonal matrix in U and computing right singular
 *              vectors of bidiagonal matrix in VT
-*              (Workspace: need M+BDSPAC)
+*              Workspace: need   3*M [e, tauq, taup] + BDSPAC
 *
                CALL DLASET( 'F', M, N, ZERO, ZERO, VT, LDVT )
                CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT,
@@ -1365,20 +1481,22 @@
 *
 *              Overwrite U by left singular vectors of A and VT
 *              by right singular vectors of A
-*              (Workspace: need 3*M, prefer 2*M+M*NB)
+*              Workspace: need   3*M [e, tauq, taup] + M    [work]
+*              Workspace: prefer 3*M [e, tauq, taup] + M*NB [work]
 *
                CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
      $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
                CALL DORMBR( 'P', 'R', 'T', M, N, M, A, LDA,
      $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
             ELSE IF( WNTQA ) THEN
 *
+*              Path 5ta (N > M, JOBZ='A')
 *              Perform bidiagonal SVD, computing left singular vectors
 *              of bidiagonal matrix in U and computing right singular
 *              vectors of bidiagonal matrix in VT
-*              (Workspace: need M+BDSPAC)
+*              Workspace: need   3*M [e, tauq, taup] + BDSPAC
 *
                CALL DLASET( 'F', N, N, ZERO, ZERO, VT, LDVT )
                CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT,
@@ -1388,20 +1506,21 @@
 *              Set the right corner of VT to identity matrix
 *
                IF( N.GT.M ) THEN
-                  CALL DLASET( 'F', N-M, N-M, ZERO, ONE, VT( M+1, M+1 ),
+                  CALL DLASET( 'F', N-M, N-M, ZERO, ONE, VT(M+1,M+1),
      $                         LDVT )
                END IF
 *
 *              Overwrite U by left singular vectors of A and VT
 *              by right singular vectors of A
-*              (Workspace: need 2*M+N, prefer 2*M+N*NB)
+*              Workspace: need   3*M [e, tauq, taup] + N    [work]
+*              Workspace: prefer 3*M [e, tauq, taup] + N*NB [work]
 *
                CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
      $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
                CALL DORMBR( 'P', 'R', 'T', N, N, M, A, LDA,
      $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
-     $                      LWORK-NWORK+1, IERR )
+     $                      LWORK - NWORK + 1, IERR )
             END IF
 *
          END IF