*DECK SODS SUBROUTINE SODS (A, X, B, NEQ, NUK, NRDA, IFLAG, WORK, IWORK) C***BEGIN PROLOGUE SODS C***SUBSIDIARY C***PURPOSE Subsidiary to BVSUP C***LIBRARY SLATEC C***TYPE SINGLE PRECISION (SODS-S) C***AUTHOR Watts, H. A., (SNLA) C***DESCRIPTION C C SODS solves the overdetermined system of linear equations A X = B, C where A is NEQ by NUK and NEQ .GE. NUK. If rank A = NUK, C X is the UNIQUE least squares solution vector. That is, C R(1)**2 + ..... + R(NEQ)**2 = minimum C where R is the residual vector R = B - A X. C If rank A .LT. NUK , the least squares solution of minimal C length can be provided. C SODS is an interfacing routine which calls subroutine LSSODS C for the solution. LSSODS in turn calls subroutine ORTHOL and C possibly subroutine OHTROR for the decomposition of A by C orthogonal transformations. In the process, ORTHOL calls upon C subroutine CSCALE 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 larger 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 WORK(*),IWORK(*) -- Arrays for storage of internal information, C WORK must be dimensioned at least 2 + 5*NUK C IWORK must be dimensioned at least NUK+2 C IWORK(2) -- 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 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 NUK C If the minimal length least squares solution is C desired, simply reset IFLAG=1 and call the code again C X -- Least squares solution of A X = B C A -- Contains the strictly upper triangular part of the reduced C matrix and the transformation information C WORK(*),IWORK(*) -- Contains information needed on subsequent C Calls (IFLAG=1 case on input) which must not C be altered C WORK(1) contains the Euclidean norm of C the residual vector C WORK(2) contains the Euclidean norm of C the solution vector C IWORK(1) contains the numerically determined C rank of the matrix A C C ********************************************************************** C C***SEE ALSO BVSUP C***REFERENCES G. Golub, Numerical methods for solving linear least C squares problems, Numerische Mathematik 7, (1965), C pp. 206-216. C P. Businger and G. Golub, Linear least squares C solutions by Householder transformations, Numerische C Mathematik 7, (1965), pp. 269-276. C H. A. Watts, Solving linear least squares problems C using SODS/SUDS/CODS, Sandia Report SAND77-0683, C Sandia Laboratories, 1977. C***ROUTINES CALLED LSSODS 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 900402 Added TYPE section. (WRB) C 910408 Updated the AUTHOR and REFERENCES sections. (WRB) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE SODS