MODULE lsq
! Module for unconstrained linear least-squares calculations.
! The algorithm is suitable for updating LS calculations as more
! data are added. This is sometimes called recursive estimation.
! Only one dependent variable is allowed.
! Based upon Applied Statistics algorithm AS 274.
! Translation from Fortran 77 to Fortran 90 by Alan Miller.
! A function, VARPRD, has been added for calculating the variances
! of predicted values, and this uses a subroutine BKSUB2.
! Version 1.14, 19 August 2002 - ELF90 compatible version
! Author: Alan Miller
! e-mail : amiller @ bigpond.net.au
! WWW-pages: http://www.ozemail.com.au/~milleraj
! http://users.bigpond.net.au/amiller/
! Bug fixes:
! 1. In REGCF a call to TOLSET has been added in case the user had
! not set tolerances.
! 2. In SING, each time a singularity is detected, unless it is in the
! variables in the last position, INCLUD is called. INCLUD assumes
! that a new observation is being added and increments the number of
! cases, NOBS. The line: nobs = nobs - 1 has been added.
! 3. row_ptr was left out of the DEALLOCATE statement in routine startup
! in version 1.07.
! 4. In COV, now calls SS if rss_set = .FALSE. 29 August 1997
! 5. In TOLSET, correction to accomodate negative values of D. 19 August 2002
! Other changes:
! 1. Array row_ptr added 18 July 1997. This points to the first element
! stored in each row thus saving a small amount of time needed to
! calculate its position.
! 2. Optional parameter, EPS, added to routine TOLSET, so that the user
! can specify the accuracy of the input data.
! 3. Cosmetic change of lsq_kind to dp (`Double precision')
! 4. Change to routine SING to use row_ptr rather than calculate the position
! of first elements in each row.
! The PUBLIC variables are:
! dp = a KIND parameter for the floating-point quantities calculated
! in this module. See the more detailed explanation below.
! This KIND parameter should be used for all floating-point
! arguments passed to routines in this module.
! nobs = the number of observations processed to date.
! ncol = the total number of variables, including one for the constant,
! if a constant is being fitted.
! r_dim = the dimension of array r = ncol*(ncol-1)/2
! vorder = an integer vector storing the current order of the variables
! in the QR-factorization. The initial order is 0, 1, 2, ...
! if a constant is being fitted, or 1, 2, ... otherwise.
! initialized = a logical variable which indicates whether space has
! been allocated for various arrays.
! tol_set = a logical variable which is set when subroutine TOLSET has
! been called to calculate tolerances for use in testing for
! singularities.
! rss_set = a logical variable indicating whether residual sums of squares
! are available and usable.
! d() = array of row multipliers for the Cholesky factorization.
! The factorization is X = Q.sqrt(D).R where Q is an ortho-
! normal matrix which is NOT stored, D is a diagonal matrix
! whose diagonal elements are stored in array d, and R is an
! upper-triangular matrix with 1's as its diagonal elements.
! rhs() = vector of RHS projections (after scaling by sqrt(D)).
! Thus Q'y = sqrt(D).rhs
! r() = the upper-triangular matrix R. The upper triangle only,
! excluding the implicit 1's on the diagonal, are stored by
! rows.
! tol() = array of tolerances used in testing for singularities.
! rss() = array of residual sums of squares. rss(i) is the residual
! sum of squares with the first i variables in the model.
! By changing the order of variables, the residual sums of
! squares can be found for all possible subsets of the variables.
! The residual sum of squares with NO variables in the model,
! that is the total sum of squares of the y-values, can be
! calculated as rss(1) + d(1)*rhs(1)^2. If the first variable
! is a constant, then rss(1) is the sum of squares of
! (y - ybar) where ybar is the average value of y.
! sserr = residual sum of squares with all of the variables included.
! row_ptr() = array of indices of first elements in each row of R.
!
!--------------------------------------------------------------------------
! General declarations
IMPLICIT NONE
INTEGER, SAVE :: nobs, ncol, r_dim
INTEGER, ALLOCATABLE, SAVE :: vorder(:), row_ptr(:)
LOGICAL, SAVE :: initialized = .false., &
tol_set = .false., rss_set = .false.
! Note. dp is being set to give at least 12 decimal digit
! representation of floating point numbers. This should be adequate
! for most problems except the fitting of polynomials. dp is
! being set so that the same code can be run on PCs and Unix systems,
! which will usually represent floating-point numbers in `double
! precision', and other systems with larger word lengths which will
! give similar accuracy in `single precision'.
INTEGER, PARAMETER :: dp = SELECTED_REAL_KIND(12,60)
REAL (dp), ALLOCATABLE, SAVE :: d(:), rhs(:), r(:), tol(:), rss(:)
REAL (dp), SAVE :: zero = 0.0_dp, one = 1.0_dp, vsmall
REAL (dp), SAVE :: sserr, toly
PUBLIC :: dp, nobs, ncol, r_dim, vorder, row_ptr, &
initialized, tol_set, rss_set, &
d, rhs, r, tol, rss, sserr
PRIVATE :: zero, one, vsmall
CONTAINS
SUBROUTINE startup(nvar, fit_const)
! Allocates dimensions for arrays and initializes to zero
! The calling program must set nvar = the number of variables, and
! fit_const = .true. if a constant is to be included in the model,
! otherwise fit_const = .false.
!
!--------------------------------------------------------------------------
IMPLICIT NONE
INTEGER, INTENT(IN) :: nvar
LOGICAL, INTENT(IN) :: fit_const
! Local variable
INTEGER :: i
vsmall = 10. * TINY(zero)
nobs = 0
IF (fit_const) THEN
ncol = nvar + 1
ELSE
ncol = nvar
END IF
IF (initialized) DEALLOCATE(d, rhs, r, tol, rss, vorder, row_ptr)
r_dim = ncol * (ncol - 1)/2
ALLOCATE( d(ncol), rhs(ncol), r(r_dim), tol(ncol), rss(ncol), vorder(ncol), &
row_ptr(ncol) )
d = zero
rhs = zero
r = zero
sserr = zero
IF (fit_const) THEN
DO i = 1, ncol
vorder(i) = i-1
END DO
ELSE
DO i = 1, ncol
vorder(i) = i
END DO
END IF ! (fit_const)
! row_ptr(i) is the position of element R(i,i+1) in array r().
row_ptr(1) = 1
DO i = 2, ncol-1
row_ptr(i) = row_ptr(i-1) + ncol - i + 1
END DO
row_ptr(ncol) = 0
initialized = .true.
tol_set = .false.
rss_set = .false.
RETURN
END SUBROUTINE startup
SUBROUTINE includ(weight, xrow, yelem)
! ALGORITHM AS75.1 APPL. STATIST. (1974) VOL.23, NO. 3
! Calling this routine updates D, R, RHS and SSERR by the
! inclusion of xrow, yelem with the specified weight.
! *** WARNING Array XROW is overwritten.
! N.B. As this routine will be called many times in most applications,
! checks have been eliminated.
!
!--------------------------------------------------------------------------
IMPLICIT NONE
REAL (dp),INTENT(IN) :: weight, yelem
REAL (dp), DIMENSION(:), INTENT(IN OUT) :: xrow
! Local variables
INTEGER :: i, k, nextr
REAL (dp) :: w, y, xi, di, wxi, dpi, cbar, sbar, xk
nobs = nobs + 1
w = weight
y = yelem
rss_set = .false.
nextr = 1
DO i = 1, ncol
! Skip unnecessary transformations. Test on exact zeroes must be
! used or stability can be destroyed.
IF (ABS(w) < vsmall) RETURN
xi = xrow(i)
IF (ABS(xi) < vsmall) THEN
nextr = nextr + ncol - i
ELSE
di = d(i)
wxi = w * xi
dpi = di + wxi*xi
cbar = di / dpi
sbar = wxi / dpi
w = cbar * w
d(i) = dpi
DO k = i+1, ncol
xk = xrow(k)
xrow(k) = xk - xi * r(nextr)
r(nextr) = cbar * r(nextr) + sbar * xk
nextr = nextr + 1
END DO
xk = y
y = xk - xi * rhs(i)
rhs(i) = cbar * rhs(i) + sbar * xk
END IF
END DO ! i = 1, ncol
! Y * SQRT(W) is now equal to the Brown, Durbin & Evans recursive
! residual.
sserr = sserr + w * y * y
RETURN
END SUBROUTINE includ
SUBROUTINE regcf(beta, nreq, ifault)
! ALGORITHM AS274 APPL. STATIST. (1992) VOL.41, NO. 2
! Modified version of AS75.4 to calculate regression coefficients
! for the first NREQ variables, given an orthogonal reduction from
! AS75.1.
!
!--------------------------------------------------------------------------
IMPLICIT NONE
INTEGER, INTENT(IN) :: nreq
INTEGER, INTENT(OUT) :: ifault
REAL (dp), DIMENSION(:), INTENT(OUT) :: beta
! Local variables
INTEGER :: i, j, nextr
! Some checks.
ifault = 0
IF (nreq < 1 .OR. nreq > ncol) ifault = ifault + 4
IF (ifault /= 0) RETURN
IF (.NOT. tol_set) CALL tolset()
DO i = nreq, 1, -1
IF (SQRT(d(i)) < tol(i)) THEN
beta(i) = zero
d(i) = zero
ifault = -i
ELSE
beta(i) = rhs(i)
nextr = row_ptr(i)
DO j = i+1, nreq
beta(i) = beta(i) - r(nextr) * beta(j)
nextr = nextr + 1
END DO ! j = i+1, nreq
END IF
END DO ! i = nreq, 1, -1
RETURN
END SUBROUTINE regcf
SUBROUTINE tolset(eps)
! ALGORITHM AS274 APPL. STATIST. (1992) VOL.41, NO. 2
! Sets up array TOL for testing for zeroes in an orthogonal
! reduction formed using AS75.1.
REAL (dp), INTENT(IN), OPTIONAL :: eps
! Unless the argument eps is set, it is assumed that the input data are
! recorded to full machine accuracy. This is often not the case.
! If, for instance, the data are recorded to `single precision' of about
! 6-7 significant decimal digits, then singularities will not be detected.
! It is suggested that in this case eps should be set equal to
! 10.0 * EPSILON(1.0)
! If the data are recorded to say 4 significant decimals, then eps should
! be set to 1.0E-03
! The above comments apply to the predictor variables, not to the
! dependent variable.
! Correction - 19 August 2002
! When negative weights are used, it is possible for an alement of D
! to be negative.
! Local variables.
!
!--------------------------------------------------------------------------
! Local variables
INTEGER :: col, row, pos
REAL (dp) :: eps1, ten = 10.0, total, work(ncol)
! EPS is a machine-dependent constant.
IF (PRESENT(eps)) THEN
eps1 = MAX(ABS(eps), ten * EPSILON(ten))
ELSE
eps1 = ten * EPSILON(ten)
END IF
! Set tol(i) = sum of absolute values in column I of R after
! scaling each element by the square root of its row multiplier,
! multiplied by EPS1.
work = SQRT(ABS(d))
DO col = 1, ncol
pos = col - 1
total = work(col)
DO row = 1, col-1
total = total + ABS(r(pos)) * work(row)
pos = pos + ncol - row - 1
END DO
tol(col) = eps1 * total
END DO
tol_set = .TRUE.
RETURN
END SUBROUTINE tolset
SUBROUTINE sing(lindep, ifault)
! ALGORITHM AS274 APPL. STATIST. (1992) VOL.41, NO. 2
! Checks for singularities, reports, and adjusts orthogonal
! reductions produced by AS75.1.
! Correction - 19 August 2002
! When negative weights are used, it is possible for an alement of D
! to be negative.
! Auxiliary routines called: INCLUD, TOLSET
!
!--------------------------------------------------------------------------
INTEGER, INTENT(OUT) :: ifault
LOGICAL, DIMENSION(:), INTENT(OUT) :: lindep
! Local variables
REAL (dp) :: temp, x(ncol), work(ncol), y, weight
INTEGER :: pos, row, pos2
ifault = 0
work = SQRT(ABS(d))
IF (.NOT. tol_set) CALL tolset()
DO row = 1, ncol
temp = tol(row)
pos = row_ptr(row) ! pos = location of first element in row
! If diagonal element is near zero, set it to zero, set appropriate
! element of LINDEP, and use INCLUD to augment the projections in
! the lower rows of the orthogonalization.
lindep(row) = .FALSE.
IF (work(row) <= temp) THEN
lindep(row) = .TRUE.
ifault = ifault - 1
IF (row < ncol) THEN
pos2 = pos + ncol - row - 1
x = zero
x(row+1:ncol) = r(pos:pos2)
y = rhs(row)
weight = d(row)
r(pos:pos2) = zero
d(row) = zero
rhs(row) = zero
CALL includ(weight, x, y)
! INCLUD automatically increases the number
! of cases each time it is called.
nobs = nobs - 1
ELSE
sserr = sserr + d(row) * rhs(row)**2
END IF ! (row < ncol)
END IF ! (work(row) <= temp)
END DO ! row = 1, ncol
RETURN
END SUBROUTINE sing
SUBROUTINE ss()
! ALGORITHM AS274 APPL. STATIST. (1992) VOL.41, NO. 2
! Calculates partial residual sums of squares from an orthogonal
! reduction from AS75.1.
!
!--------------------------------------------------------------------------
! Local variables
INTEGER :: i
REAL (dp) :: total
total = sserr
rss(ncol) = sserr
DO i = ncol, 2, -1
total = total + d(i) * rhs(i)**2
rss(i-1) = total
END DO
rss_set = .TRUE.
RETURN
END SUBROUTINE ss
SUBROUTINE cov(nreq, var, covmat, dimcov, sterr, ifault)
! ALGORITHM AS274 APPL. STATIST. (1992) VOL.41, NO. 2
! Calculate covariance matrix for regression coefficients for the
! first nreq variables, from an orthogonal reduction produced from
! AS75.1.
! Auxiliary routine called: INV
!
!--------------------------------------------------------------------------
INTEGER, INTENT(IN) :: nreq, dimcov
INTEGER, INTENT(OUT) :: ifault
REAL (dp), INTENT(OUT) :: var
REAL (dp), DIMENSION(:), INTENT(OUT) :: covmat, sterr
! Local variables.
INTEGER :: dim_rinv, pos, row, start, pos2, col, pos1, k
REAL (dp) :: total
REAL (dp), ALLOCATABLE :: rinv(:)
! Check that dimension of array covmat is adequate.
IF (dimcov < nreq*(nreq+1)/2) THEN
ifault = 1
RETURN
END IF
! Check for small or zero multipliers on the diagonal.
ifault = 0
DO row = 1, nreq
IF (ABS(d(row)) < vsmall) ifault = -row
END DO
IF (ifault /= 0) RETURN
! Calculate estimate of the residual variance.
IF (nobs > nreq) THEN
IF (.NOT. rss_set) CALL ss()
var = rss(nreq) / (nobs - nreq)
ELSE
ifault = 2
RETURN
END IF
dim_rinv = nreq*(nreq-1)/2
ALLOCATE ( rinv(dim_rinv) )
CALL INV(nreq, rinv)
pos = 1
start = 1
DO row = 1, nreq
pos2 = start
DO col = row, nreq
pos1 = start + col - row
IF (row == col) THEN
total = one / d(col)
ELSE
total = rinv(pos1-1) / d(col)
END IF
DO K = col+1, nreq
total = total + rinv(pos1) * rinv(pos2) / d(k)
pos1 = pos1 + 1
pos2 = pos2 + 1
END DO ! K = col+1, nreq
covmat(pos) = total * var
IF (row == col) sterr(row) = SQRT(covmat(pos))
pos = pos + 1
END DO ! col = row, nreq
start = start + nreq - row
END DO ! row = 1, nreq
DEALLOCATE(rinv)
RETURN
END SUBROUTINE cov
SUBROUTINE inv(nreq, rinv)
! ALGORITHM AS274 APPL. STATIST. (1992) VOL.41, NO. 2
! Invert first nreq rows and columns of Cholesky factorization
! produced by AS 75.1.
!
!--------------------------------------------------------------------------
INTEGER, INTENT(IN) :: nreq
REAL (dp), DIMENSION(:), INTENT(OUT) :: rinv
! Local variables.
INTEGER :: pos, row, col, start, k, pos1, pos2
REAL (dp) :: total
! Invert R ignoring row multipliers, from the bottom up.
pos = nreq * (nreq-1)/2
DO row = nreq-1, 1, -1
start = row_ptr(row)
DO col = nreq, row+1, -1
pos1 = start
pos2 = pos
total = zero
DO k = row+1, col-1
pos2 = pos2 + nreq - k
total = total - r(pos1) * rinv(pos2)
pos1 = pos1 + 1
END DO ! k = row+1, col-1
rinv(pos) = total - r(pos1)
pos = pos - 1
END DO ! col = nreq, row+1, -1
END DO ! row = nreq-1, 1, -1
RETURN
END SUBROUTINE inv
SUBROUTINE partial_corr(in, cormat, dimc, ycorr, ifault)
! Replaces subroutines PCORR and COR of:
! ALGORITHM AS274 APPL. STATIST. (1992) VOL.41, NO. 2
! Calculate partial correlations after the variables in rows
! 1, 2, ..., IN have been forced into the regression.
! If IN = 1, and the first row of R represents a constant in the
! model, then the usual simple correlations are returned.
! If IN = 0, the value returned in array CORMAT for the correlation
! of variables Xi & Xj is:
! sum ( Xi.Xj ) / Sqrt ( sum (Xi^2) . sum (Xj^2) )
! On return, array CORMAT contains the upper triangle of the matrix of
! partial correlations stored by rows, excluding the 1's on the diagonal.
! e.g. if IN = 2, the consecutive elements returned are:
! (3,4) (3,5) ... (3,ncol), (4,5) (4,6) ... (4,ncol), etc.
! Array YCORR stores the partial correlations with the Y-variable
! starting with YCORR(IN+1) = partial correlation with the variable in
! position (IN+1).
!
!--------------------------------------------------------------------------
INTEGER, INTENT(IN) :: in, dimc
INTEGER, INTENT(OUT) :: ifault
REAL (dp), DIMENSION(:), INTENT(OUT) :: cormat, ycorr
! Local variables.
INTEGER :: base_pos, pos, row, col, col1, col2, pos1, pos2
REAL (dp) :: rms(in+1:ncol), sumxx, sumxy, sumyy, work(in+1:ncol)
! Some checks.
ifault = 0
IF (in < 0 .OR. in > ncol-1) ifault = ifault + 4
IF (dimc < (ncol-in)*(ncol-in-1)/2) ifault = ifault + 8
IF (ifault /= 0) RETURN
! Base position for calculating positions of elements in row (IN+1) of R.
base_pos = in*ncol - (in+1)*(in+2)/2
! Calculate 1/RMS of elements in columns from IN to (ncol-1).
IF (d(in+1) > zero) rms(in+1) = one / SQRT(d(in+1))
DO col = in+2, ncol
pos = base_pos + col
sumxx = d(col)
DO row = in+1, col-1
sumxx = sumxx + d(row) * r(pos)**2
pos = pos + ncol - row - 1
END DO ! row = in+1, col-1
IF (sumxx > zero) THEN
rms(col) = one / SQRT(sumxx)
ELSE
rms(col) = zero
ifault = -col
END IF ! (sumxx > zero)
END DO ! col = in+1, ncol-1
! Calculate 1/RMS for the Y-variable
sumyy = sserr
DO row = in+1, ncol
sumyy = sumyy + d(row) * rhs(row)**2
END DO ! row = in+1, ncol
IF (sumyy > zero) sumyy = one / SQRT(sumyy)
! Calculate sums of cross-products.
! These are obtained by taking dot products of pairs of columns of R,
! but with the product for each row multiplied by the row multiplier
! in array D.
pos = 1
DO col1 = in+1, ncol
sumxy = zero
work(col1+1:ncol) = zero
pos1 = base_pos + col1
DO row = in+1, col1-1
pos2 = pos1 + 1
DO col2 = col1+1, ncol
work(col2) = work(col2) + d(row) * r(pos1) * r(pos2)
pos2 = pos2 + 1
END DO ! col2 = col1+1, ncol
sumxy = sumxy + d(row) * r(pos1) * rhs(row)
pos1 = pos1 + ncol - row - 1
END DO ! row = in+1, col1-1
! Row COL1 has an implicit 1 as its first element (in column COL1)
pos2 = pos1 + 1
DO col2 = col1+1, ncol
work(col2) = work(col2) + d(col1) * r(pos2)
pos2 = pos2 + 1
cormat(pos) = work(col2) * rms(col1) * rms(col2)
pos = pos + 1
END DO ! col2 = col1+1, ncol
sumxy = sumxy + d(col1) * rhs(col1)
ycorr(col1) = sumxy * rms(col1) * sumyy
END DO ! col1 = in+1, ncol-1
ycorr(1:in) = zero
RETURN
END SUBROUTINE partial_corr
SUBROUTINE vmove(from, to, ifault)
! ALGORITHM AS274 APPL. STATIST. (1992) VOL.41, NO. 2
! Move variable from position FROM to position TO in an
! orthogonal reduction produced by AS75.1.
!
!--------------------------------------------------------------------------
INTEGER, INTENT(IN) :: from, to
INTEGER, INTENT(OUT) :: ifault
! Local variables
REAL (dp) :: d1, d2, x, d1new, d2new, cbar, sbar, y
INTEGER :: m, first, last, inc, m1, m2, mp1, col, pos, row
! Check input parameters
ifault = 0
IF (from < 1 .OR. from > ncol) ifault = ifault + 4
IF (to < 1 .OR. to > ncol) ifault = ifault + 8
IF (ifault /= 0) RETURN
IF (from == to) RETURN
IF (.NOT. rss_set) CALL ss()
IF (from < to) THEN
first = from
last = to - 1
inc = 1
ELSE
first = from - 1
last = to
inc = -1
END IF
DO m = first, last, inc
! Find addresses of first elements of R in rows M and (M+1).
m1 = row_ptr(m)
m2 = row_ptr(m+1)
mp1 = m + 1
d1 = d(m)
d2 = d(mp1)
! Special cases.
IF (d1 < vsmall .AND. d2 < vsmall) GO TO 40
x = r(m1)
IF (ABS(x) * SQRT(d1) < tol(mp1)) THEN
x = zero
END IF
IF (d1 < vsmall .OR. ABS(x) < vsmall) THEN
d(m) = d2
d(mp1) = d1
r(m1) = zero
DO col = m+2, ncol
m1 = m1 + 1
x = r(m1)
r(m1) = r(m2)
r(m2) = x
m2 = m2 + 1
END DO ! col = m+2, ncol
x = rhs(m)
rhs(m) = rhs(mp1)
rhs(mp1) = x
GO TO 40
ELSE IF (d2 < vsmall) THEN
d(m) = d1 * x**2
r(m1) = one / x
r(m1+1:m1+ncol-m-1) = r(m1+1:m1+ncol-m-1) / x
rhs(m) = rhs(m) / x
GO TO 40
END IF
! Planar rotation in regular case.
d1new = d2 + d1*x**2
cbar = d2 / d1new
sbar = x * d1 / d1new
d2new = d1 * cbar
d(m) = d1new
d(mp1) = d2new
r(m1) = sbar
DO col = m+2, ncol
m1 = m1 + 1
y = r(m1)
r(m1) = cbar*r(m2) + sbar*y
r(m2) = y - x*r(m2)
m2 = m2 + 1
END DO ! col = m+2, ncol
y = rhs(m)
rhs(m) = cbar*rhs(mp1) + sbar*y
rhs(mp1) = y - x*rhs(mp1)
! Swap columns M and (M+1) down to row (M-1).
40 pos = m
DO row = 1, m-1
x = r(pos)
r(pos) = r(pos-1)
r(pos-1) = x
pos = pos + ncol - row - 1
END DO ! row = 1, m-1
! Adjust variable order (VORDER), the tolerances (TOL) and
! the vector of residual sums of squares (RSS).
m1 = vorder(m)
vorder(m) = vorder(mp1)
vorder(mp1) = m1
x = tol(m)
tol(m) = tol(mp1)
tol(mp1) = x
rss(m) = rss(mp1) + d(mp1) * rhs(mp1)**2
END DO
RETURN
END SUBROUTINE vmove
SUBROUTINE reordr(list, n, pos1, ifault)
! ALGORITHM AS274 APPL. STATIST. (1992) VOL.41, NO. 2
! Re-order the variables in an orthogonal reduction produced by
! AS75.1 so that the N variables in LIST start at position POS1,
! though will not necessarily be in the same order as in LIST.
! Any variables in VORDER before position POS1 are not moved.
! Auxiliary routine called: VMOVE
!
!--------------------------------------------------------------------------
INTEGER, INTENT(IN) :: n, pos1
INTEGER, DIMENSION(:), INTENT(IN) :: list
INTEGER, INTENT(OUT) :: ifault
! Local variables.
INTEGER :: next, i, l, j
! Check N.
ifault = 0
IF (n < 1 .OR. n > ncol+1-pos1) ifault = ifault + 4
IF (ifault /= 0) RETURN
! Work through VORDER finding variables which are in LIST.
next = pos1
i = pos1
10 l = vorder(i)
DO j = 1, n
IF (l == list(j)) GO TO 40
END DO
30 i = i + 1
IF (i <= ncol) GO TO 10
! If this point is reached, one or more variables in LIST has not
! been found.
ifault = 8
RETURN
! Variable L is in LIST; move it up to position NEXT if it is not
! already there.
40 IF (i > next) CALL vmove(i, next, ifault)
next = next + 1
IF (next < n+pos1) GO TO 30
RETURN
END SUBROUTINE reordr
SUBROUTINE hdiag(xrow, nreq, hii, ifault)
! ALGORITHM AS274 APPL. STATIST. (1992) VOL.41, NO. 2
!
! -1 -1
! The hat matrix H = x(X'X) x' = x(R'DR) x' = z'Dz
! -1
! where z = x'R
! Here we only calculate the diagonal element hii corresponding to one
! row (xrow). The variance of the i-th least-squares residual is (1 - hii).
!--------------------------------------------------------------------------
INTEGER, INTENT(IN) :: nreq
INTEGER, INTENT(OUT) :: ifault
REAL (dp), DIMENSION(:), INTENT(IN) :: xrow
REAL (dp), INTENT(OUT) :: hii
! Local variables
INTEGER :: col, row, pos
REAL (dp) :: total, wk(ncol)
! Some checks
ifault = 0
IF (nreq > ncol) ifault = ifault + 4
IF (ifault /= 0) RETURN
! The elements of xrow.inv(R).sqrt(D) are calculated and stored in WK.
hii = zero
DO col = 1, nreq
IF (SQRT(d(col)) <= tol(col)) THEN
wk(col) = zero
ELSE
pos = col - 1
total = xrow(col)
DO row = 1, col-1
total = total - wk(row)*r(pos)
pos = pos + ncol - row - 1
END DO ! row = 1, col-1
wk(col) = total
hii = hii + total**2 / d(col)
END IF
END DO ! col = 1, nreq
RETURN
END SUBROUTINE hdiag
FUNCTION varprd(x, nreq) RESULT(fn_val)
! Calculate the variance of x'b where b consists of the first nreq
! least-squares regression coefficients.
!
!--------------------------------------------------------------------------
INTEGER, INTENT(IN) :: nreq
REAL (dp), DIMENSION(:), INTENT(IN) :: x
REAL (dp) :: fn_val
! Local variables
INTEGER :: ifault, row
REAL (dp) :: var, wk(nreq)
! Check input parameter values
fn_val = zero
ifault = 0
IF (nreq < 1 .OR. nreq > ncol) ifault = ifault + 4
IF (nobs <= nreq) ifault = ifault + 8
IF (ifault /= 0) THEN
WRITE(*, '(1x, a, i4)') 'Error in function VARPRD: ifault =', ifault
RETURN
END IF
! Calculate the residual variance estimate.
var = sserr / (nobs - nreq)
! Variance of x'b = var.x'(inv R)(inv D)(inv R')x
! First call BKSUB2 to calculate (inv R')x by back-substitution.
CALL BKSUB2(x, wk, nreq)
DO row = 1, nreq
IF(d(row) > tol(row)) fn_val = fn_val + wk(row)**2 / d(row)
END DO
fn_val = fn_val * var
RETURN
END FUNCTION varprd
SUBROUTINE bksub2(x, b, nreq)
! Solve x = R'b for b given x, using only the first nreq rows and
! columns of R, and only the first nreq elements of R.
!
!--------------------------------------------------------------------------
INTEGER, INTENT(IN) :: nreq
REAL (dp), DIMENSION(:), INTENT(IN) :: x
REAL (dp), DIMENSION(:), INTENT(OUT) :: b
! Local variables
INTEGER :: pos, row, col
REAL (dp) :: temp
! Solve by back-substitution, starting from the top.
DO row = 1, nreq
pos = row - 1
temp = x(row)
DO col = 1, row-1
temp = temp - r(pos)*b(col)
pos = pos + ncol - col - 1
END DO
b(row) = temp
END DO
RETURN
END SUBROUTINE bksub2
END MODULE lsq