MODULE Gauss_Weights ! ALGORITHM 125, COLLECTED ALGORITHMS FROM ACM. ! THIS WORK PUBLISHED IN COMMUNICATIONS OF THE ACM, ! VOL. 5, NO. 10, October, 1962, P. 510--511 ! Code converted using TO_F90 by Alan Miller ! Date: 2003-06-03 Time: 14:03:32 ! Original Fortran translation donated by ! M.Dow@anu.edu.au ! ANUSF, Australian National University ! Canberra, Australia ! Tidied up to use workspace arrays, dimension arrays to ! required lengths, use 0: for dimensions of A, add comments, ! and NAG tools to layout source ! trh (20/07/97) IMPLICIT NONE INTEGER, PARAMETER :: dp = SELECTED_REAL_KIND(14, 60) CONTAINS SUBROUTINE weightc(n, q, e, eps, w, x) ! Computes the abscissae x(i) and the weight coefficients w(i) for a ! Gaussian quadrature method ! \int_0^b w(x)f(x) dx \approx \sum_{i=1}^{n}w_if(x_i) where ! \int_0^b w(x) dx = 1 and w(x)>= 0. The method requires the order n, a ! tolerance eps and the 2n-1 first coefficients of the continued fraction ! \int_0^b {w(x) \over z-x} = { 1| \over |z} - {q_1 | \over |1} - ! {e_1 | \over |z} - {q_2 | \over |1} - ! {e_2 | \over |z} - \cdots ! to be given, the latter in the two arrays q(n) and e(n-1) all components ! of which are automatically positive by virtue of the condition w(x)>= 0. ! The method works as well if the upper bound b is actually infinity ! (note that b does not appear directly as an argument!) or if the density ! function w(x) dx is replaced by da(x) with a monotonically increasing a(x) ! with at least n points of of variation. The tolerance eps should be given ! in accordance to the machines accuracy (preferably by using the value of ! d1mach(4)). The result is delivered as two arrays w(n) (the weight ! coefficients) and x(n) (the abscissae). For a description of the method see ! H Rutishauser, On a modification of the QD-algorithm with Graeffe-type ! convergence'' [Proceedings of the IFIPS Congress, Munich, 1962]. INTEGER, INTENT(IN) :: n REAL (dp), INTENT(IN) :: q(n) REAL (dp), INTENT(IN) :: e(n-1) REAL (dp), INTENT(IN OUT) :: eps REAL (dp), INTENT(OUT) :: w(n) REAL (dp), INTENT(OUT) :: x(n) ! Allocate workspaces G & A REAL (dp) :: g(n), a(0:n,0:7) ! .. ! .. Local Scalars .. REAL (dp) :: m, p INTEGER :: k ! .. ! .. Intrinsic Functions .. ! INTRINSIC ABS,EXP,LOG ! .. x(1) = q(1) + e(1) DO k = 2,n g(k-1) = e(k-1)*q(k)/x(k-1) IF (k == n) THEN x(k) = q(k) - g(k-1) ELSE x(k) = q(k) + e(k) - g(k-1) END IF g(k-1) = g(k-1)/x(k) w(k-1) = x(k)/x(k-1) x(k-1) = LOG(x(k-1)) END DO x(n) = LOG(x(n)) p = 1 30 DO k = 1,n - 1 IF (ABS(g(k)*w(k)) > eps) GO TO 40 END DO GO TO 50 40 CALL qdgraeffe(n, x, g, w, a) p = 2*p GO TO 30 ! What follows is a peculiar method to compute the w(k) from ! the given ratios g_k = w_{k+1}/w_k such that ! \sum_{k=1}^n w_k = 1, but the straightforward formulae to do ! this might well produce overflow of exponent 50 w(1) = 1 m = 0 DO k = 1,n - 1 w(k+1) = w(k)*g(k) IF (w(k) > m) m = w(k) END DO ! WRITE (6,FMT=*) 'm=',m m = SUM( w(1:n) ) DO k = 1,n w(k) = w(k)/m x(k) = EXP(x(k)/p) END DO RETURN END SUBROUTINE weightc SUBROUTINE red(a, f, n) ! Subroutine RED reduces a heptadiagonal matrix a to tridiagonal form as ! described in the paper referenced above. INTEGER, INTENT(IN) :: n REAL (dp), INTENT(IN OUT) :: a(0:n,0:7) REAL (dp), INTENT(IN) :: f(n) ! .. REAL (dp) :: c INTEGER :: j, k ! .. ! .. Intrinsic Functions .. ! INTRINSIC ABS ! .. DO k = 1,n - 1 DO j = k,n - 1 c = -f(j)*a(j,7)/a(j,2) a(j,7) = 0 a(j+1,2) = a(j+1,2) + c*a(j,5) a(j,1) = a(j,1) - c*f(j)*a(j,4) a(j,6) = a(j,6) - c*a(j+1,1) a(j+1,3) = a(j+1,3) - c*a(j+1,6) END DO DO j = k,n - 1 c = -f(j)*a(j,4)/a(j,1) a(j,4) = 0 a(j+1,1) = a(j+1,1) + c*a(j,6) a(j+1,6) = a(j+1,6) + c*a(j+1,3) a(j,5) = a(j,5) - c*a(j+1,2) a(j+1,0) = a(j+1,0) - c*a(j+1,5) END DO DO j = k + 1,n - 1 c = -a(j,3)/a(j-1,6) a(j,3) = 0 a(j,6) = a(j,6) + c*a(j,1) a(j-1,5) = a(j-1,5) - c*f(j)*f(j)*a(j,0) a(j,2) = a(j,2) - c*f(j)*f(j)*a(j,5) a(j,7) = a(j,7) - c*f(j)*a(j+1,2) END DO DO j = k + 1,n - 1 c = -a(j,0)/a(j-1,5) a(j,0) = 0 a(j+1,2) = a(j+1,2) + c*f(j)*a(j,7) a(j,5) = a(j,5) + c*a(j,2) a(j,1) = a(j,1) - c*f(j)*f(j)*a(j,6) a(j,4) = a(j,4) - c*f(j)*a(j+1,1) END DO END DO RETURN END SUBROUTINE red SUBROUTINE qdgraeffe(n, h, g, f, a) ! Subroutine QDGRAEFFE computes for a given continued fraction ! f(z) = { 1| \over |z} - {q_1 | \over |1} - ! {e_1 | \over |z} - {q_2 | \over |1} - ! {e_2 | \over |z} - \cdots - {q_n | \over |1} ! another one, the poles of which are the squares of the poles of f(z) ! However QDGRAEFFE uses not the coefficients q_1 ... q_n and ! e_1 ... e_{n-1} of f(z) but the quotients f_k = q_{k+1}/q_k and ! g_k = e_k/q_{k+1} for k=1,2,...,n-1 and the h_k = ln(abs(q_k)) for ! k=1,2,...,n, and the results are delivered in the same form. Routine ! QDGRAEFFE can be used independently, but requires subroutine RED INTEGER, INTENT(IN) :: n REAL (dp), INTENT(OUT) :: h(n) REAL (dp), INTENT(OUT) :: g(n) REAL (dp), INTENT(OUT) :: f(n) REAL (dp), INTENT(OUT) :: a(0:n,0:7) ! .. ! .. Local Scalars .. INTEGER :: k ! .. ! .. External Subroutines .. ! EXTERNAL red ! .. g(n) = 0 f(n) = 0 DO k = 1,n a(k-1,4) = 1.0_dp a(k-1,5) = 1.0_dp a(k,2) = 1 + g(k)*f(k) a(k,1) = 1 + g(k)*f(k) a(k,6) = g(k) a(k,7) = g(k) a(k,0) = 0.0_dp a(k,3) = 0.0_dp END DO a(n,5) = 0.0_dp ! The array a represents the heptadiagonal matrix Q of the paper ! cited above, but with the modifications needed to avoid large numbers ! and with a peculiar arrangement. CALL red(a, f, n) DO k = 1,n h(k) = 2*h(k) + LOG(ABS(a(k,1)*a(k,2))) END DO DO k = 1,n - 1 f(k) = f(k)*f(k)*a(k+1,2)*a(k+1,1)/ (a(k,1)*a(k,2)) g(k) = a(k,5)*a(k,6)/ (a(k+1,1)*a(k+1,2)) END DO RETURN END SUBROUTINE qdgraeffe END MODULE Gauss_Weights PROGRAM main ! Code converted using TO_F90 by Alan Miller ! Date: 2003-06-03 Time: 14:01:08 ! Driver for CACM Alg 125 Rutishauser ! Author M.Dow@anu.edu.au ! ANUSF, Australian National University ! Canberra, Australia ! Tidied up to use workspace arrays, real parameter values ! and put through nag tools ! trh (20/07/97) USE Gauss_Weights IMPLICIT NONE ! .. ! .. Parameters .. INTEGER, PARAMETER :: nm = 100 REAL (dp), PARAMETER :: zero=0.0D0, one=1.0D0, two=2.0D0, three=3.0D0 ! .. Local Scalars .. REAL (dp) :: a, b, eps, exact, s INTEGER :: i, n, p ! .. ! .. Local Arrays .. REAL (dp) :: e(nm), q(nm), w(nm), x(nm) ! .. ! .. External Subroutines .. ! EXTERNAL weightcoeff ! .. 10 WRITE (6, FMT=*) ' Enter n, eps: ' READ (5, FMT=*, END=20) n, eps IF (eps <= zero) eps = 1.D-15 IF (n >= nm-1) STOP WRITE (6, FMT=*) 'n=', n, ' eps=', eps ! q,e for w=1/2 interval (0,2) ! Ref: W.Jones & W.Thron Continued Fractions ... ! Encyclopedia of maths...vol 11 p24 ! Continued Fraction expansion of log(1+x), ! transformed to Rutishauser form p33 DO i = 2,nm q(i) = two*i*i/ (two*i* (two*i-one)) e(i) = two*i*i/ (two*i* (two*i+one)) END DO q(1) = one e(1) = one/three CALL weightc(n, q, e, eps, w, x) ! Adjust weights, zeros for w=1, interval (-1,1) DO i = 1, n w(i) = two*w(i) x(i) = x(i) - one END DO DO i = 1, MIN(n,10) WRITE (6, FMT=9000) w(i), x(i) END DO ! Check for x^4 p = 4 a = -one b = one exact = (b** (p+1)-a** (p+1))/ (p+1) s = zero DO i = 1, n s = s + w(i)*x(i)**p END DO WRITE (6, FMT=9010) p, exact, s, exact - s GO TO 10 20 STOP 9000 FORMAT (f20.16, ' ', f20.16) 9010 FORMAT (i3, ' Exact=', f20.16, ' Quadrature=', f20.16, ' Error=', e14.7) END PROGRAM main