*DECK LSSUDS SUBROUTINE LSSUDS (A, X, B, N, M, NRDA, U, NRDU, IFLAG, MLSO, + IRANK, ISCALE, Q, DIAG, KPIVOT, S, DIV, TD, ISFLG, SCALES) C***BEGIN PROLOGUE LSSUDS C***SUBSIDIARY C***PURPOSE Subsidiary to BVSUP C***LIBRARY SLATEC C***TYPE SINGLE PRECISION (LSSUDS-S, DLSSUD-D) C***AUTHOR Watts, H. A., (SNLA) C***DESCRIPTION C C LSSUDS solves the underdetermined system of equations A Z = B, C where A is N by M and N .LE. M. In particular, if rank A equals C IRA, a vector X and a matrix U are determined such that X is the C UNIQUE solution of smallest length, satisfying A X = B, and the C columns of U form an orthonormal basis for the null space of A, C satisfying A U = 0 . Then all solutions Z are given by C Z = X + C(1)*U(1) + ..... + C(M-IRA)*U(M-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 C ********************************************************************* C INPUT C ********************************************************************* C C A -- Contains the matrix of N equations in M unknowns, A remains C unchanged, must be dimensioned NRDA by M. C X -- Solution array of length at least M. C B -- Given constant vector of length N, B remains unchanged. C N -- Number of equations, N greater or equal to 1. C M -- Number of unknowns, M greater or equal to N. C NRDA -- Row dimension of A, NRDA greater or equal to N. C U -- Matrix used for solution, must be dimensioned NRDU by C (M - rank of A). C (storage for U may be ignored when only the minimal length C solution X is desired) C NRDU -- Row dimension of U, NRDU greater or equal to M. C (if only the minimal length solution is wanted, C NRDU=0 is acceptable) 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. C IRANK -- Variable used for the rank of A, set by the code. C ISCALE -- Scaling indicator C =-1 if the matrix A 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 Q -- Matrix used for the transformation, must be dimensioned C NRDA by M. C DIAG,KPIVOT,S, -- Arrays of length at least N used for internal C DIV,TD,SCALES storage (except for SCALES which is M). C ISFLG -- Storage for an internal variable. 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 N. C To continue, simply reset IFLAG=1 and call LSSUDS 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 Z = B C IRANK -- Numerically determined rank of A, must not be altered C on succeeding calls with input values of IFLAG=1. C U -- Matrix whose M-IRANK columns are mutually orthogonal unit C vectors which span the null space of A. This is to be ignored C when MLSO was set to zero or IFLAG=4 on output. C Q -- Contains the strictly upper triangular part of the reduced C matrix and transformation information. C DIAG -- Contains the diagonal elements of the triangular reduced C matrix. C KPIVOT -- Contains the pivotal information. The row interchanges C performed on the original matrix are recorded here. C S -- Contains the solution of the lower triangular system. C DIV,TD -- Contains transformation information for rank C deficient problems. C SCALES -- Contains the column scaling parameters. C C ********************************************************************* C C***SEE ALSO BVSUP 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 J4SAVE, OHTROL, ORTHOR, R1MACH, SDOT, XERMAX, C XERMSG, XGETF, XSETF C***REVISION HISTORY (YYMMDD) C 750601 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890831 Modified array declarations. (WRB) C 891214 Prologue converted to Version 4.0 format. (BAB) C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ) C 900328 Added TYPE section. (WRB) C 900510 Fixed an error message. (RWC) C 910408 Updated the AUTHOR and REFERENCES sections. (WRB) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE LSSUDS