SLATEC Routines --- WNNLS ---
*DECK WNNLS
SUBROUTINE WNNLS (W, MDW, ME, MA, N, L, PRGOPT, X, RNORM, MODE,
+ IWORK, WORK)
C***BEGIN PROLOGUE WNNLS
C***PURPOSE Solve a linearly constrained least squares problem with
C equality constraints and nonnegativity constraints on
C selected variables.
C***LIBRARY SLATEC
C***CATEGORY K1A2A
C***TYPE SINGLE PRECISION (WNNLS-S, DWNNLS-D)
C***KEYWORDS CONSTRAINED LEAST SQUARES, CURVE FITTING, DATA FITTING,
C EQUALITY CONSTRAINTS, INEQUALITY CONSTRAINTS,
C NONNEGATIVITY CONSTRAINTS, QUADRATIC PROGRAMMING
C***AUTHOR Hanson, R. J., (SNLA)
C Haskell, K. H., (SNLA)
C***DESCRIPTION
C
C Abstract
C
C This subprogram solves a linearly constrained least squares
C problem. Suppose there are given matrices E and A of
C respective dimensions ME by N and MA by N, and vectors F
C and B of respective lengths ME and MA. This subroutine
C solves the problem
C
C EX = F, (equations to be exactly satisfied)
C
C AX = B, (equations to be approximately satisfied,
C in the least squares sense)
C
C subject to components L+1,...,N nonnegative
C
C Any values ME.GE.0, MA.GE.0 and 0.LE. L .LE.N are permitted.
C
C The problem is reposed as problem WNNLS
C
C (WT*E)X = (WT*F)
C ( A) ( B), (least squares)
C subject to components L+1,...,N nonnegative.
C
C The subprogram chooses the heavy weight (or penalty parameter) WT.
C
C The parameters for WNNLS are
C
C INPUT..
C
C W(*,*),MDW, The array W(*,*) is double subscripted with first
C ME,MA,N,L dimensioning parameter equal to MDW. For this
C discussion let us call M = ME + MA. Then MDW
C must satisfy MDW.GE.M. The condition MDW.LT.M
C is an error.
C
C The array W(*,*) contains the matrices and vectors
C
C (E F)
C (A B)
C
C in rows and columns 1,...,M and 1,...,N+1
C respectively. Columns 1,...,L correspond to
C unconstrained variables X(1),...,X(L). The
C remaining variables are constrained to be
C nonnegative. The condition L.LT.0 or L.GT.N is
C an error.
C
C PRGOPT(*) This real-valued array is the option vector.
C If the user is satisfied with the nominal
C subprogram features set
C
C PRGOPT(1)=1 (or PRGOPT(1)=1.0)
C
C Otherwise PRGOPT(*) is a linked list consisting of
C groups of data of the following form
C
C LINK
C KEY
C DATA SET
C
C The parameters LINK and KEY are each one word.
C The DATA SET can be comprised of several words.
C The number of items depends on the value of KEY.
C The value of LINK points to the first
C entry of the next group of data within
C PRGOPT(*). The exception is when there are
C no more options to change. In that
C case LINK=1 and the values KEY and DATA SET
C are not referenced. The general layout of
C PRGOPT(*) is as follows.
C
C ...PRGOPT(1)=LINK1 (link to first entry of next group)
C . PRGOPT(2)=KEY1 (key to the option change)
C . PRGOPT(3)=DATA VALUE (data value for this change)
C . .
C . .
C . .
C ...PRGOPT(LINK1)=LINK2 (link to the first entry of
C . next group)
C . PRGOPT(LINK1+1)=KEY2 (key to the option change)
C . PRGOPT(LINK1+2)=DATA VALUE
C ... .
C . .
C . .
C ...PRGOPT(LINK)=1 (no more options to change)
C
C Values of LINK that are nonpositive are errors.
C A value of LINK.GT.NLINK=100000 is also an error.
C This helps prevent using invalid but positive
C values of LINK that will probably extend
C beyond the program limits of PRGOPT(*).
C Unrecognized values of KEY are ignored. The
C order of the options is arbitrary and any number
C of options can be changed with the following
C restriction. To prevent cycling in the
C processing of the option array a count of the
C number of options changed is maintained.
C Whenever this count exceeds NOPT=1000 an error
C message is printed and the subprogram returns.
C
C OPTIONS..
C
C KEY=6
C Scale the nonzero columns of the
C entire data matrix
C (E)
C (A)
C to have length one. The DATA SET for
C this option is a single value. It must
C be nonzero if unit length column scaling is
C desired.
C
C KEY=7
C Scale columns of the entire data matrix
C (E)
C (A)
C with a user-provided diagonal matrix.
C The DATA SET for this option consists
C of the N diagonal scaling factors, one for
C each matrix column.
C
C KEY=8
C Change the rank determination tolerance from
C the nominal value of SQRT(SRELPR). This quantity
C can be no smaller than SRELPR, The arithmetic-
C storage precision. The quantity used
C here is internally restricted to be at
C least SRELPR. The DATA SET for this option
C is the new tolerance.
C
C KEY=9
C Change the blow-up parameter from the
C nominal value of SQRT(SRELPR). The reciprocal of
C this parameter is used in rejecting solution
C components as too large when a variable is
C first brought into the active set. Too large
C means that the proposed component times the
C reciprocal of the parameter is not less than
C the ratio of the norms of the right-side
C vector and the data matrix.
C This parameter can be no smaller than SRELPR,
C the arithmetic-storage precision.
C
C For example, suppose we want to provide
C a diagonal matrix to scale the problem
C matrix and change the tolerance used for
C determining linear dependence of dropped col
C vectors. For these options the dimensions of
C PRGOPT(*) must be at least N+6. The FORTRAN
C statements defining these options would
C be as follows.
C
C PRGOPT(1)=N+3 (link to entry N+3 in PRGOPT(*))
C PRGOPT(2)=7 (user-provided scaling key)
C
C CALL SCOPY(N,D,1,PRGOPT(3),1) (copy the N
C scaling factors from a user array called D(*)
C into PRGOPT(3)-PRGOPT(N+2))
C
C PRGOPT(N+3)=N+6 (link to entry N+6 of PRGOPT(*))
C PRGOPT(N+4)=8 (linear dependence tolerance key)
C PRGOPT(N+5)=... (new value of the tolerance)
C
C PRGOPT(N+6)=1 (no more options to change)
C
C
C IWORK(1), The amounts of working storage actually allocated
C IWORK(2) for the working arrays WORK(*) and IWORK(*),
C respectively. These quantities are compared with
C the actual amounts of storage needed for WNNLS( ).
C Insufficient storage allocated for either WORK(*)
C or IWORK(*) is considered an error. This feature
C was included in WNNLS( ) because miscalculating
C the storage formulas for WORK(*) and IWORK(*)
C might very well lead to subtle and hard-to-find
C execution errors.
C
C The length of WORK(*) must be at least
C
C LW = ME+MA+5*N
C This test will not be made if IWORK(1).LE.0.
C
C The length of IWORK(*) must be at least
C
C LIW = ME+MA+N
C This test will not be made if IWORK(2).LE.0.
C
C OUTPUT..
C
C X(*) An array dimensioned at least N, which will
C contain the N components of the solution vector
C on output.
C
C RNORM The residual norm of the solution. The value of
C RNORM contains the residual vector length of the
C equality constraints and least squares equations.
C
C MODE The value of MODE indicates the success or failure
C of the subprogram.
C
C MODE = 0 Subprogram completed successfully.
C
C = 1 Max. number of iterations (equal to
C 3*(N-L)) exceeded. Nearly all problems
C should complete in fewer than this
C number of iterations. An approximate
C solution and its corresponding residual
C vector length are in X(*) and RNORM.
C
C = 2 Usage error occurred. The offending
C condition is noted with the error
C processing subprogram, XERMSG( ).
C
C User-designated
C Working arrays..
C
C WORK(*) A real-valued working array of length at least
C M + 5*N.
C
C IWORK(*) An integer-valued working array of length at least
C M+N.
C
C***REFERENCES K. H. Haskell and R. J. Hanson, An algorithm for
C linear least squares problems with equality and
C nonnegativity constraints, Report SAND77-0552, Sandia
C Laboratories, June 1978.
C K. H. Haskell and R. J. Hanson, Selected algorithms for
C the linearly constrained least squares problem - a
C users guide, Report SAND78-1290, Sandia Laboratories,
C August 1979.
C K. H. Haskell and R. J. Hanson, An algorithm for
C linear least squares problems with equality and
C nonnegativity constraints, Mathematical Programming
C 21 (1981), pp. 98-118.
C R. J. Hanson and K. H. Haskell, Two algorithms for the
C linearly constrained least squares problem, ACM
C Transactions on Mathematical Software, September 1982.
C C. L. Lawson and R. J. Hanson, Solving Least Squares
C Problems, Prentice-Hall, Inc., 1974.
C***ROUTINES CALLED WNLSM, XERMSG
C***REVISION HISTORY (YYMMDD)
C 790701 DATE WRITTEN
C 890206 REVISION DATE from Version 3.2
C 890618 Completely restructured and revised. (WRB & RWC)
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ)
C 900510 Convert XERRWV calls to XERMSG calls. (RWC)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE WNNLS