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+.help svdsol Jun99 "Slalib Package"
+.nf
+
+      SUBROUTINE slSVDS (M, N, MP, NP, B, U, W, V, WORK, X)
+
+     - - - - - - -
+      S V D S
+     - - - - - - -
+
+  From a given vector and the SVD of a matrix (as obtained from
+  the SVD routine), obtain the solution vector (double precision)
+
+  This routine solves the equation:
+
+     A . x = b
+
+  where:
+
+     A   is a given M (rows) x N (columns) matrix, where M.GE.N
+     x   is the N-vector we wish to find
+     b   is a given M-vector
+
+  by means of the Singular Value Decomposition method (SVD).  In
+  this method, the matrix A is first factorised (for example by
+  the routine slSVD) into the following components:
+
+     A = U x W x VT
+
+  where:
+
+     A   is the M (rows) x N (columns) matrix
+     U   is an M x N column-orthogonal matrix
+     W   is an N x N diagonal matrix with W(I,I).GE.0
+     VT  is the transpose of an NxN orthogonal matrix
+
+     Note that M and N, above, are the LOGICAL dimensions of the
+     matrices and vectors concerned, which can be located in
+     arrays of larger PHYSICAL dimensions MP and NP.
+
+  The solution is found from the expression:
+
+     x = V . [diag(1/Wj)] . (transpose(U) . b)
+
+  Notes:
+
+  1)  If matrix A is square, and if the diagonal matrix W is not
+      adjusted, the method is equivalent to conventional solution
+      of simultaneous equations.
+
+  2)  If M>N, the result is a least-squares fit.
+
+  3)  If the solution is poorly determined, this shows up in the
+      SVD factorisation as very small or zero Wj values.  Where
+      a Wj value is small but non-zero it can be set to zero to
+      avoid ill effects.  The present routine detects such zero
+      Wj values and produces a sensible solution, with highly
+      correlated terms kept under control rather than being allowed
+      to elope to infinity, and with meaningful values for the
+      other terms.
+
+  Given:
+     M,N    i         numbers of rows and columns in matrix A
+     MP,NP  i         physical dimensions of array containing matrix A
+     B      d(M)      known vector b
+     U      d(MP,NP)  array containing MxN matrix U
+     W      d(N)      NxN diagonal matrix W (diagonal elements only)
+     V      d(NP,NP)  array containing NxN orthogonal matrix V
+
+  Returned:
+     WORK   d(N)      workspace
+     X      d(N)      unknown vector x
+
+  Reference:
+     Numerical Recipes, section 2.9.
+
+  P.T.Wallace   Starlink   29 October 1993
+
+  Copyright (C) 1995 Rutherford Appleton Laboratory
+  Copyright (C) 1995 Association of Universities for Research in Astronomy Inc.
+
+.fi
+.endhelp