SLATEC Routines --- HFTI ---
*DECK HFTI
SUBROUTINE HFTI (A, MDA, M, N, B, MDB, NB, TAU, KRANK, RNORM, H,
+ G, IP)
C***BEGIN PROLOGUE HFTI
C***PURPOSE Solve a linear least squares problems by performing a QR
C factorization of the matrix using Householder
C transformations.
C***LIBRARY SLATEC
C***CATEGORY D9
C***TYPE SINGLE PRECISION (HFTI-S, DHFTI-D)
C***KEYWORDS CURVE FITTING, LINEAR LEAST SQUARES, QR FACTORIZATION
C***AUTHOR Lawson, C. L., (JPL)
C Hanson, R. J., (SNLA)
C***DESCRIPTION
C
C DIMENSION A(MDA,N),(B(MDB,NB) or B(M)),RNORM(NB),H(N),G(N),IP(N)
C
C This subroutine solves a linear least squares problem or a set of
C linear least squares problems having the same matrix but different
C right-side vectors. The problem data consists of an M by N matrix
C A, an M by NB matrix B, and an absolute tolerance parameter TAU
C whose usage is described below. The NB column vectors of B
C represent right-side vectors for NB distinct linear least squares
C problems.
C
C This set of problems can also be written as the matrix least
C squares problem
C
C AX = B,
C
C where X is the N by NB solution matrix.
C
C Note that if B is the M by M identity matrix, then X will be the
C pseudo-inverse of A.
C
C This subroutine first transforms the augmented matrix (A B) to a
C matrix (R C) using premultiplying Householder transformations with
C column interchanges. All subdiagonal elements in the matrix R are
C zero and its diagonal elements satisfy
C
C ABS(R(I,I)).GE.ABS(R(I+1,I+1)),
C
C I = 1,...,L-1, where
C
C L = MIN(M,N).
C
C The subroutine will compute an integer, KRANK, equal to the number
C of diagonal terms of R that exceed TAU in magnitude. Then a
C solution of minimum Euclidean length is computed using the first
C KRANK rows of (R C).
C
C To be specific we suggest that the user consider an easily
C computable matrix norm, such as, the maximum of all column sums of
C magnitudes.
C
C Now if the relative uncertainty of B is EPS, (norm of uncertainty/
C norm of B), it is suggested that TAU be set approximately equal to
C EPS*(norm of A).
C
C The user must dimension all arrays appearing in the call list..
C A(MDA,N),(B(MDB,NB) or B(M)),RNORM(NB),H(N),G(N),IP(N). This
C permits the solution of a range of problems in the same array
C space.
C
C The entire set of parameters for HFTI are
C
C INPUT..
C
C A(*,*),MDA,M,N The array A(*,*) initially contains the M by N
C matrix A of the least squares problem AX = B.
C The first dimensioning parameter of the array
C A(*,*) is MDA, which must satisfy MDA.GE.M
C Either M.GE.N or M.LT.N is permitted. There
C is no restriction on the rank of A. The
C condition MDA.LT.M is considered an error.
C
C B(*),MDB,NB If NB = 0 the subroutine will perform the
C orthogonal decomposition but will make no
C references to the array B(*). If NB.GT.0
C the array B(*) must initially contain the M by
C NB matrix B of the least squares problem AX =
C B. If NB.GE.2 the array B(*) must be doubly
C subscripted with first dimensioning parameter
C MDB.GE.MAX(M,N). If NB = 1 the array B(*) may
C be either doubly or singly subscripted. In
C the latter case the value of MDB is arbitrary
C but it should be set to some valid integer
C value such as MDB = M.
C
C The condition of NB.GT.1.AND.MDB.LT. MAX(M,N)
C is considered an error.
C
C TAU Absolute tolerance parameter provided by user
C for pseudorank determination.
C
C H(*),G(*),IP(*) Arrays of working space used by HFTI.
C
C OUTPUT..
C
C A(*,*) The contents of the array A(*,*) will be
C modified by the subroutine. These contents
C are not generally required by the user.
C
C B(*) On return the array B(*) will contain the N by
C NB solution matrix X.
C
C KRANK Set by the subroutine to indicate the
C pseudorank of A.
C
C RNORM(*) On return, RNORM(J) will contain the Euclidean
C norm of the residual vector for the problem
C defined by the J-th column vector of the array
C B(*,*) for J = 1,...,NB.
C
C H(*),G(*) On return these arrays respectively contain
C elements of the pre- and post-multiplying
C Householder transformations used to compute
C the minimum Euclidean length solution.
C
C IP(*) Array in which the subroutine records indices
C describing the permutation of column vectors.
C The contents of arrays H(*),G(*) and IP(*)
C are not generally required by the user.
C
C***REFERENCES C. L. Lawson and R. J. Hanson, Solving Least Squares
C Problems, Prentice-Hall, Inc., 1974, Chapter 14.
C***ROUTINES CALLED H12, R1MACH, XERMSG
C***REVISION HISTORY (YYMMDD)
C 790101 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 891006 Cosmetic changes to prologue. (WRB)
C 891006 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ)
C 901005 Replace usage of DIFF with usage of R1MACH. (RWC)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE HFTI