jnl.c revision ddc0e0b53c661f6e439e3b7072b3ef353eadb4af
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25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Use is subject to license terms.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * floating point Bessel's function of the 1st and 2nd kind
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * of order n: jn(n,x),yn(n,x);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Special cases:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Note 2. About jn(n,x), yn(n,x)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * For n=0, j0(x) is called,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * for n=1, j1(x) is called,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * for n<x, forward recursion us used starting
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * from values of j0(x) and j1(x).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * for n>x, a continued fraction approximation to
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * j(n,x)/j(n-1,x) is evaluated and then backward
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * recursion is used starting from a supposed value
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * for j(n,x). The resulting value of j(0,x) is
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * compared with the actual value to correct the
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * supposed value of j(n,x).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * yn(n,x) is similar in all respects, except
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * that forward recursion is used for all
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * values of n>1.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis#define GENERIC long double
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtisinvsqrtpi = 5.641895835477562869480794515607725858441e-0001L,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Thus, J(-n,x) = J(n,-x)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if ((n&1) == 0)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis else if ((GENERIC)n <= x) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (x > 1.0e91L) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Let s=sin(x), c=cos(x),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * n sin(xn)*sqt2 cos(xn)*sqt2
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ----------------------------------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis switch (n&3) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis for (i = 1; i < n; i++) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis b = b*((GENERIC)(i+i)/x) - a; /* avoid underflow */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (x < 1e-17L) { /* use J(n,x) = 1/n!*(x/2)^n */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis for (a = one, i = 1; i <= n; i++) a *= (GENERIC)i;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* use backward recurrence */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * J(n,x)/J(n-1,x) = ---- ------ ------ .....
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 2n - 2(n+1) - 2(n+2)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (for large x) = ---- ------ ------ .....
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 2n 2(n+1) 2(n+2)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * -- - ------ - ------ -
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Let w = 2n/x and h=2/x, then the above quotient
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * is equal to the continued fraction:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = -----------------------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * w - -----------------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * w+h - ---------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * To determine how many terms needed, let
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Q(0) = w, Q(1) = w(w+h) - 1,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * When Q(k) > 1e4 good for single
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * When Q(k) > 1e9 good for double
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * When Q(k) > 1e17 good for quaduple
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* determin k */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis w = (n+n)/(double)x; h = 2.0/(double)x;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis k += 1; z += h;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis for (t = zero, i = 2*(n+k); i >= m; i -= 2) t = one/(i/x-t);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * hence, if n*(log(2n/x)) > ...
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * single 8.8722839355e+01
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * double 7.09782712893383973096e+02
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * long double 1.1356523406294143949491931077970765006170e+04
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * then recurrent value may overflow and the result is
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * likely underflow to zero
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (tmp < 1.1356523406294143949491931077970765e+04L) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis for (i = n-1; i > 0; i--) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis b = ((i+i)/x)*b - a;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis for (i = n-1; i > 0; i--) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis b = ((i+i)/x)*b - a;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (b > 1e1000L) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis b = (t*j0l(x)/b);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis Let s = sin(x), c = cos(x),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis xn = x-(2n+1)*pi/4, sqt2 = sqrt(2), then
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis n sin(xn)*sqt2 cos(xn)*sqt2
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis ----------------------------------
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis switch (n&3) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * fix 1262058 and take care of non-default rounding
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis for (i = 1; i < n; i++) {
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis b *= (GENERIC) (i + i) / x;