*DECK DSUDS SUBROUTINE DSUDS (A, X, B, NEQ, NUK, NRDA, IFLAG, MLSO, WORK, + IWORK) C***BEGIN PROLOGUE DSUDS C***SUBSIDIARY C***PURPOSE Subsidiary to DBVSUP C***LIBRARY SLATEC C***TYPE DOUBLE PRECISION (SUDS-S, DSUDS-D) C***AUTHOR Watts, H. A., (SNLA) C***DESCRIPTION C C DSUDS solves the underdetermined system of linear equations A Z = C B where A is NEQ by NUK and NEQ .LE. NUK. in particular, if rank C A equals IRA, a vector X and a matrix U are determined such that C X is the UNIQUE solution of smallest length, satisfying A X = B, C and the columns of U form an orthonormal basis for the null C space of A, satisfying A U = 0 . Then all solutions Z are C given by C Z = X + C(1)*U(1) + ..... + C(NUK-IRA)*U(NUK-IRA) C where U(J) represents the J-th column of U and the C(J) are C arbitrary constants. C If the system of equations are not compatible, only the least C squares solution of minimal length is computed. C DSUDS is an interfacing routine which calls subroutine DLSSUD C for the solution. DLSSUD in turn calls subroutine DORTHR and C possibly subroutine DOHTRL for the decomposition of A by C orthogonal transformations. In the process, DORTHR calls upon C subroutine DCSCAL for scaling. C C ******************************************************************** C INPUT C ******************************************************************** C C A -- Contains the matrix of NEQ equations in NUK unknowns and must C be dimensioned NRDA by NUK. The original A is destroyed. C X -- Solution array of length at least NUK. C B -- Given constant vector of length NEQ, B is destroyed. C NEQ -- Number of equations, NEQ greater or equal to 1. C NUK -- Number of columns in the matrix (which is also the number C of unknowns), NUK not smaller than NEQ. C NRDA -- Row dimension of A, NRDA greater or equal to NEQ. C IFLAG -- Status indicator C =0 for the first call (and for each new problem defined by C a new matrix A) when the matrix data is treated as exact C =-K for the first call (and for each new problem defined by C a new matrix A) when the matrix data is assumed to be C accurate to about K digits. C =1 for subsequent calls whenever the matrix A has already C been decomposed (problems with new vectors B but C same matrix A can be handled efficiently). C MLSO -- =0 if only the minimal length solution is wanted. C =1 if the complete solution is wanted, includes the C linear space defined by the matrix U in the abstract. C WORK(*),IWORK(*) -- Arrays for storage of internal information, C WORK must be dimensioned at least C NUK + 3*NEQ + MLSO*NUK*(NUK-RANK A) C where it is possible for 0 .LE. RANK A .LE. NEQ C IWORK must be dimensioned at least 3 + NEQ C IWORK(2) -- Scaling indicator C =-1 if the matrix is to be pre-scaled by C columns when appropriate. C If the scaling indicator is not equal to -1 C no scaling will be attempted. C For most problems scaling will probably not be necessary C C ********************************************************************* C OUTPUT C ********************************************************************* C C IFLAG -- Status indicator C =1 if solution was obtained. C =2 if improper input is detected. C =3 if rank of matrix is less than NEQ. C to continue simply reset IFLAG=1 and call DSUDS again. C =4 if the system of equations appears to be inconsistent. C However, the least squares solution of minimal length C was obtained. C X -- Minimal length least squares solution of A X = B. C A -- Contains the strictly upper triangular part of the reduced C matrix and transformation information. C WORK(*),IWORK(*) -- Contains information needed on subsequent C calls (IFLAG=1 case on input) which must not C be altered. C The matrix U described in the abstract is C stored in the NUK*(NUK-rank A) elements of C the WORK array beginning at WORK(1+NUK+3*NEQ). C However U is not defined when MLSO=0 or C IFLAG=4. C IWORK(1) contains the numerically determined C rank of the matrix A C C ********************************************************************* C C***SEE ALSO DBVSUP C***REFERENCES H. A. Watts, Solving linear least squares problems C using SODS/SUDS/CODS, Sandia Report SAND77-0683, C Sandia Laboratories, 1977. C***ROUTINES CALLED DLSSUD C***REVISION HISTORY (YYMMDD) C 750601 DATE WRITTEN C 890831 Modified array declarations. (WRB) C 891214 Prologue converted to Version 4.0 format. (BAB) C 900328 Added TYPE section. (WRB) C 910408 Updated the AUTHOR and REFERENCES sections. (WRB) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE DSUDS