/*-------------------------------------------------------------------- * $Id: gmt_stat.c,v 1.58 2007/09/28 19:49:43 remko Exp $ * * Copyright (c) 1991-2007 by P. Wessel and W. H. F. Smith * See COPYING file for copying and redistribution conditions. * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation; version 2 of the License. * * This program is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * Contact info: gmt.soest.hawaii.edu *--------------------------------------------------------------------*/ /* * some stuff for significance tests. * * Author: Walter H. F. Smith * Date: 12-JUN-1995 * Version: 4.1.x * Revised: 12-AUG-1998 * 06-JAN-1999 P Wessel: Added GMT_ber, GMT_bei, GMT_ker, GMT_kei * 12-JAN-1999 P Wessel: Added GMT_plm * 14-JAN-1999 P Wessel: Added GMT_erf, GMT_erfc * 20-JAN-1999 P Wessel: Added GMT_i0, GMT_k0 * 26-JAN-1999 P Wessel: Added GMT_i1, GMT_k1, GMT_in, GMT_kn * 06-MAY-1999 P Wessel: Added GMT_dilog * 18-JUN-1999 W Smith: Changed GMT_ber, GMT_bei, GMT_ker, GMT_kei * to speed up telescoped power series, and * include asymptotic series for x > 8. * 21-JUN-1999 W Smith: Revised GMT_plm and GMT_factorial. * 12-AUG-1999 W Smith: Revised GMT_sig_f() to use new routines; * added GMT_f_q() and GMT_student_t_a(); * added GMT_f_test_new(). * 10-SEP-1999 P Wessel: Added GMT_erfinv * 18-OCT-1999 P Wessel: Added sincos when not already defined * Added GMT_j0, GMT_j1, GMT_jn, GMT_y0, * GMT_y1, GMT_yn for platforms that do no * have the bessel functions in the math lib. * 07-DEC-1999 P. Wessel: Added GMT_rand and GMT_nrand * 29-MAR-2000 W Smith: Added GMT_hypot, GMT_log1p, and GMT_atanh for * platforms without native library support. * 09-MAY-2000 P. Wessel: Added GMT_chi2 and GMT_cumpoisson * 18-AUG-2000 P. Wessel: Moved GMT_mode and GMT_median from gmt_support to here. * Added float versions of these two functions for grd data. * 27-JUL-2005 P. Wessel: Added Chebyshev polynomials Tn(x) * 07-SEP-2005 P. Wessel: Added GMT_corrcoeff (x,y) * 24-MAR-2006 P. Wessel: Added GMT_zdist (x) * 06-JUL-2007 P. Wessel: Added GMT_psi () and GMT_PvQv() * 28-SEP-2007 R. Scharroo: Added GMT_plm_bar and made GMT_factorial public * * PUBLIC functions: * * GMT_f_test : Routine to compute the probability that two variances are the same * GMT_f_test_new: As above, but allows choosing 1- or 2-sided, and which side * GMT_f_q: Returns the probability integral Q(F,nu1,nu2) of the F-distribution. * GMT_student_t_a: Returns the prob integral A(t,nu) of the student-t distrib. * GMT_sig_f : Returns TRUE if reduction in model misfit was significant * GMT_bei: Kelvin-Bessel function bei(x) * GMT_ber: Kelvin-Bessel function ber(x) * GMT_kei: Kelvin-Bessel function kei(x) * GMT_ker: Kelvin-Bessel function ker(x) * GMT_plm: Legendre polynomial of degree L order M * GMT_i0: Modified Bessel function 1st kind order 0 * GMT_i1: Modified Bessel function 1st kind order 1 * GMT_i2: Modified Bessel function 1st kind order N * GMT_k0: Modified Kelvin function 2nd kind order 0 * GMT_k1: Modified Kelvin function 2nd kind order 1 * GMT_kn: Modified Kelvin function 2nd kind order N * GMT_j0: Bessel function 1st kind order 0 * GMT_j1: Bessel function 1st kind order 1 * GMT_jn: Bessel function 1st kind order N * GMT_y0: Bessel function 2nd kind order 0 * GMT_y1: Bessel function 2nd kind order 1 * GMT_yn: Bessel function 2nd kind order N * GMT_dilog: The dilog function * GMT_erfinv: The inverse error function * GMT_rand: Uniformly distributed random numbers 0 < x < 1 * GMT_nrand: Normally distributed random numbers from N(0,1) * GMT_corrcoeff: Correlation coefficient. * GMT_psi: Digamma (psi) function. * GMT_PvQv: Legendre functions Pv and Qv for imaginary v and real x (-1/+1). * GMT_factorial: Factorials. */ #define GMT_WITH_NO_PS #include "gmt.h" int GMT_inc_beta (double a, double b, double x, double *ibeta); int GMT_ln_gamma_r (double x, double *lngam); int GMT_f_test_new (double chisq1, int nu1, double chisq2, int nu2, double *prob, int iside); int GMT_student_t_a (double t, int n, double *prob); double GMT_ln_gamma (double xx); double GMT_cf_beta (double a, double b, double x); double GMT_bei (double x); double GMT_kei (double x); double GMT_ber (double x); double GMT_ker (double x); double GMT_plm (int l, int m, double x); double GMT_factorial (int n); double GMT_i0 (double x); double GMT_i1 (double x); double GMT_in (int n, double x); double GMT_k0 (double x); double GMT_k1 (double x); double GMT_kn (int n, double x); double GMT_dilog (double x); double GMT_ln_gamma (double xx); /* Computes natural log of the gamma function */ double GMT_erfinv (double y); double GMT_gammq (double a, double x); void GMT_gamma_cf (double *gammcf, double a, double x, double *gln); void GMT_gamma_ser (double *gamser, double a, double x, double *gln); void GMT_cumpoisson (double k, double mu, double *prob); #if HAVE_J0 == 0 double GMT_j0 (double x); #endif #if HAVE_J1 == 0 double GMT_j1 (double x); #endif #if HAVE_JN == 0 double GMT_jn (int n, double x); #endif #if HAVE_Y0 == 0 double GMT_y0 (double x); #endif #if HAVE_Y1 == 0 double GMT_y1 (double x); #endif #if HAVE_YN == 0 double GMT_yn (int n, double x); #endif #if HAVE_SINCOS == 0 && HAVE_ALPHASINCOS == 0 void sincos (double a, double *s, double *c); #endif int GMT_f_test_new (double chisq1, int nu1, double chisq2, int nu2, double *prob, int iside) { /* Given chisq1 and chisq2, random variables distributed as chi-square with nu1 and nu2 degrees of freedom, respectively, except that chisq1 is scaled by var1, and chisq2 is scaled by var2, let the null hypothesis, H0, be that var1 = var2. This routine assigns prob, the probability that we can reject H0 in favor of a new hypothesis, H1, according to iside: iside=+1 means H1 is that var1 > var2 iside=-1 means H1 is that var1 < var2 iside=0 means H1 is that var1 != var2. This routine differs from the old GMT_f_test() by adding the argument iside and allowing one to choose the test. The old routine in effect always set iside=0. This routine also differs from GMT_f_test() in that the former used the incomplete beta function and this one uses GMT_f_q(). Returns 0 on success, -1 on failure. WHF Smith, 12 August 1999. */ double q; /* The probability from GMT_f_q(), which is the prob that H0 should be retained even though chisq1/nu1 > chisq2/nu2. */ if (chisq1 <= 0.0 || chisq2 <= 0.0 || nu1 < 1 || nu2 < 1) { *prob = GMT_d_NaN; fprintf (stderr, "GMT_f_test_new: ERROR: Bad argument(s).\n"); return (-1); } GMT_f_q (chisq1, nu1, chisq2, nu2, &q); if (iside > 0) { *prob = 1.0 - q; } else if (iside < 0) { *prob = q; } else if ( (chisq1/nu1) <= (chisq2/nu2) ) { *prob = 2.0*q; } else { *prob = 2.0*(1.0 - q); } return (0); } int GMT_f_test (double chisq1, int nu1, double chisq2, int nu2, double *prob) { /* Routine to compute the probability that two variances are the same. chisq1 is distributed as chisq with nu1 degrees of freedom; ditto for chisq2 and nu2. If these are independent and we form the ratio F = max(chisq1,chisq2)/min(chisq1,chisq2) then we can ask what is the probability that an F greater than this would occur by chance. It is this probability that is returned in prob. When prob is small, it is likely that the two chisq represent two different populations; the confidence that the two do not represent the same pop is 1.0 - prob. This is a two-sided test. This follows some ideas in Numerical Recipes, CRC Handbook, and Abramowitz and Stegun. */ double f, df1, df2, p1, p2; if ( chisq1 <= 0.0) { fprintf(stderr,"GMT_f_test: Chi-Square One <= 0.0\n"); return(-1); } if ( chisq2 <= 0.0) { fprintf(stderr,"GMT_f_test: Chi-Square Two <= 0.0\n"); return(-1); } if (chisq1 > chisq2) { f = chisq1/chisq2; df1 = nu1; df2 = nu2; } else { f = chisq2/chisq1; df1 = nu2; df2 = nu1; } if (GMT_inc_beta(0.5*df2, 0.5*df1, df2/(df2+df1*f), &p1) ) { fprintf(stderr,"GMT_f_test: Trouble on 1st GMT_inc_beta call.\n"); return(-1); } if (GMT_inc_beta(0.5*df1, 0.5*df2, df1/(df1+df2/f), &p2) ) { fprintf(stderr,"GMT_f_test: Trouble on 2nd GMT_inc_beta call.\n"); return(-1); } *prob = p1 + (1.0 - p2); return(0); } int GMT_sig_f (double chi1, int n1, double chi2, int n2, double level, double *prob) { /* Returns TRUE if chi1/n1 significantly less than chi2/n2 at the level level. Returns FALSE if: error occurs in GMT_f_test_new(); chi1/n1 not significantly < chi2/n2 at level. Changed 12 August 1999 to use GMT_f_test_new() */ int trouble; trouble = GMT_f_test_new (chi1, n1, chi2, n2, prob, -1); if (trouble) return(0); return((*prob) >= level); } /* --------- LOWER LEVEL FUNCTIONS ------- */ int GMT_inc_beta (double a, double b, double x, double *ibeta) { double bt, gama, gamb, gamab; if (a <= 0.0) { fprintf(stderr,"GMT_inc_beta: Bad a (a <= 0).\n"); return(-1); } if (b <= 0.0) { fprintf(stderr,"GMT_inc_beta: Bad b (b <= 0).\n"); return(-1); } if (x > 0.0 && x < 1.0) { GMT_ln_gamma_r(a, &gama); GMT_ln_gamma_r(b, &gamb); GMT_ln_gamma_r( (a+b), &gamab); bt = exp(gamab - gama - gamb + a * d_log(x) + b * d_log(1.0 - x) ); /* Here there is disagreement on the range of x which converges efficiently. Abramowitz and Stegun say to use x < (a - 1) / (a + b - 2). Editions of Numerical Recipes thru mid 1987 say x < ( (a + 1) / (a + b + 1), but the code has x < ( (a + 1) / (a + b + 2). Editions printed late 1987 and after say x < ( (a + 1) / (a + b + 2) in text as well as code. What to do ? */ if (x < ( (a + 1) / (a + b + 2) ) ) { *ibeta = bt * GMT_cf_beta(a, b, x) / a; } else { *ibeta = 1.0 - bt * GMT_cf_beta(b, a, (1.0 - x) ) / b; } return(0); } else if (x == 0.0) { *ibeta = 0.0; return(0); } else if (x == 1.0) { *ibeta = 1.0; return(0); } else if (x < 0.0) { fprintf(stderr,"GMT_inc_beta: Bad x (x < 0).\n"); *ibeta = 0.0; } else if (x > 1.0) { fprintf(stderr,"GMT_inc_beta: Bad x (x > 1).\n"); *ibeta = 1.0; } return(-1); } int GMT_ln_gamma_r(double x, double *lngam) { /* Get natural logrithm of Gamma(x), x > 0. To maintain full accuracy, this routine uses Gamma(1 + x) / x when x < 1. This routine in turn calls GMT_ln_gamma(x), which computes the actual function value. GMT_ln_gamma assumes it is being called in a smart way, and does not check the range of x. */ if (x > 1.0) { *lngam = GMT_ln_gamma(x); return(0); } if (x > 0.0 && x < 1.0) { *lngam = GMT_ln_gamma(1.0 + x) - d_log(x); return(0); } if (x == 1.0) { *lngam = 0.0; return(0); } fprintf(stderr,"Ln Gamma: Bad x (x <= 0).\n"); return(-1); } double GMT_ln_gamma (double xx) { /* Routine to compute natural log of Gamma(x) by Lanczos approximation. Most accurate for x > 1; fails for x <= 0. No error checking is done here; it is assumed that this is called by GMT_ln_gamma_r() */ static double cof[6] = { 76.18009173, -86.50532033, 24.01409822, -1.231739516, 0.120858003e-2, -0.536382e-5 }; static double stp = 2.50662827465, half = 0.5, one = 1.0, fpf = 5.5; double x, tmp, ser; int i; x = xx - one; tmp = x + fpf; tmp = (x + half) * d_log(tmp) - tmp; ser = one; for (i = 0; i < 6; i++) { x += one; ser += (cof[i]/x); } return(tmp + d_log(stp*ser) ); } double GMT_cf_beta (double a, double b, double x) { /* Continued fraction method called by GMT_inc_beta. */ static int itmax = 100; static double eps = 3.0e-7; double am = 1.0, bm = 1.0, az = 1.0; double qab, qap, qam, bz, em, tem, d; double ap, bp, app, bpp, aold; int m = 0; qab = a + b; qap = a + 1.0; qam = a - 1.0; bz = 1.0 - qab * x / qap; do { m++; em = m; tem = em + em; d = em*(b-m)*x/((qam+tem)*(a+tem)); ap = az+d*am; bp = bz+d*bm; d = -(a+m)*(qab+em)*x/((a+tem)*(qap+tem)); app = ap+d*az; bpp = bp+d*bz; aold = az; am = ap/bpp; bm = bp/bpp; az = app/bpp; bz = 1.0; } while ( ( (fabs(az-aold) ) >= (eps * fabs(az) ) ) && (m < itmax) ); if (m == itmax) fprintf(stderr,"GMT_cf_beta: A or B too big, or ITMAX too small.\n"); return(az); } #define ITMAX 100 void GMT_gamma_ser (double *gamser, double a, double x, double *gln) { /* Returns the incomplete gamma function P(a,x) by series rep. * Press et al, gser() */ int n; double sum, del, ap; GMT_ln_gamma_r (a, gln); if (x < 0.0) { fprintf (stderr, "GMT DOMAIN ERROR: x < 0 in GMT_gamma_ser(x)\n"); *gamser = GMT_d_NaN; return; } if (x == 0.0) { *gamser = 0.0; return; } ap = a; del = sum = 1.0 / a; for (n = 1; n <= ITMAX; n++) { ap += 1.0; del *= x / ap; sum += del; if (fabs (del) < fabs (sum) * DBL_EPSILON) { *gamser = sum * exp (-x + a * log (x) - (*gln)); return; } } fprintf (stderr, "GMT DOMAIN ERROR: a too large, ITMAX too small in GMT_gamma_ser(x)\n"); } void GMT_gamma_cf (double *gammcf, double a, double x, double *gln) { /* Returns the incomplete gamma function P(a,x) by continued fraction. * Press et al, gcf() */ int n; double gold = 0.0, g, fac = 1.0, b1 = 1.0; double b0 = 0.0, anf, ana, an, a1, a0 = 1.0; GMT_ln_gamma_r (a, gln); a1 = x; for (n = 1; n <= ITMAX; n++) { an = (double) n; ana = an - a; a0 = (a1 + a0 * ana) * fac; b0 = (b1 + b0 * ana) * fac; anf = an * fac; a1 = x * a0 + anf * a1; b1 = x * b0 + anf * b1; if (a1 != 0.0) { fac = 1.0 / a1; g = b1 * fac; if (fabs ((g - gold) / g) < DBL_EPSILON) { *gammcf = exp (-x + a * log (x) - (*gln)) * g; return; } gold = g; } } fprintf (stderr, "GMT DOMAIN ERROR: a too large, ITMAX too small in GMT_gamma_cf(x)\n"); } double GMT_gammq (double a, double x) { /* Returns Q(a,x) = 1 - P(a,x) Inc. Gamma function */ double G, gln; if (x < 0.0 || a <= 0.0) { fprintf (stderr, "GMT DOMAIN ERROR: Invalid arguments to GMT_gammaq\n"); return (GMT_d_NaN); } if (x < (a + 1.0)) { GMT_gamma_ser (&G, a, x, &gln); return (1.0 - G); } GMT_gamma_cf (&G, a, x, &gln); return (G); } /* * Kelvin-Bessel functions, ber, bei, ker, kei. ber(x) and bei(x) are even; * ker(x) and kei(x) are defined only for x > 0 and x >=0, respectively. * For x <= 8 we use polynomial approximations in Abramowitz & Stegun. * For x > 8 we use asymptotic series of Russell, quoted in Watson (Theory * of Bessel Functions). */ double GMT_ber (double x) { double t, rxsq, alpha, beta; if (x == 0.0) return (1.0); /* ber is an even function of x: */ x = fabs(x); if (x <= 8.0) { /* Telescoped power series from Abramowitz & Stegun */ t = x * 0.125; t *= t; t *= t; /* t = pow(x/8, 4) */ return (1.0 + t*(-64.0 + t*( 113.77777774 + t*( -32.36345652 + t*( 2.64191397 + t*( -0.08349609 + t*( 0.00122552 - 0.00000901 * t))))))); } else { /* Russell's asymptotic approximation, from Watson, p. 204 */ rxsq = 1.0 / (x * x); t = x / M_SQRT2; alpha = t; beta = t - 0.125 * M_PI; t *= 0.125 * rxsq; alpha += t; beta -= t; beta -= 0.0625*rxsq; t *= (25.0/48.0)*rxsq; alpha -= t; beta -= t; alpha -= (13.0/128.0)*(rxsq*rxsq); return ( exp(alpha) * cos(beta) / sqrt(2.0 * M_PI * x) ); } } double GMT_bei (double x) { double t, rxsq, alpha, beta; if (x == 0.0) return (0.0); /* bei is an even function of x: */ x = fabs(x); if (x <= 8.0) { /* Telescoped power series from Abramowitz & Stegun */ t = x * 0.125; rxsq = t*t; t = rxsq * rxsq; /* t = pow(x/8, 4) */ return (rxsq * (16.0 + t*( -113.77777774 + t*( 72.81777742 + t*( -10.56765779 + t*( 0.52185615 + t*( -0.01103667 + t*( 0.00011346)))))))); } else { /* Russell's asymptotic approximation, from Watson, p. 204 */ rxsq = 1.0 / (x * x); t = x / M_SQRT2; alpha = t; beta = t - 0.125 * M_PI; t *= 0.125 * rxsq; alpha += t; beta -= t; beta -= 0.0625*rxsq; t *= (25.0/48.0)*rxsq; alpha -= t; beta -= t; alpha -= (13.0/128.0)*(rxsq*rxsq); return ( exp(alpha) * sin(beta) / sqrt(2.0 * M_PI * x) ); } } double GMT_ker (double x) { double t, rxsq, alpha, beta; if (x <= 0.0) { fprintf (stderr, "GMT DOMAIN ERROR: x <= 0 in GMT_ker(x)\n"); return (GMT_d_NaN); } if (x <= 8.0) { /* Telescoped power series from Abramowitz & Stegun */ t = 0.125 * x; t *= t; t *= t; /* t = pow(x/8, 4) */ return (-log(0.5 * x) * GMT_ber(x) + 0.25 * M_PI * GMT_bei(x) -M_EULER + t * ( -59.05819744 + t * ( 171.36272133 + t * ( -60.60977451 + t * ( 5.65539121 + t * ( -0.199636347 + t * ( 0.00309699 + t * ( -0.00002458 * t)))))))); } else { /* Russell's asymptotic approximation, from Watson, p. 204 */ rxsq = 1.0 / (x * x); t = -x / M_SQRT2; alpha = t; beta = t - 0.125 * M_PI; t *= 0.125 * rxsq; alpha += t; beta -= t; beta -= 0.0625*rxsq; t *= (25.0/48.0)*rxsq; alpha -= t; beta -= t; alpha -= (13.0/128.0)*(rxsq*rxsq); return ( exp(alpha) * cos(beta) / sqrt(2.0 * x / M_PI) ); } } double GMT_kei (double x) { double t, rxsq, alpha, beta; if (x <= 0.0) { /* Zero is valid. If near enough to zero, return kei(0) */ if (x > -GMT_CONV_LIMIT) return (-0.25 * M_PI); fprintf (stderr, "GMT DOMAIN ERROR: x < 0 in GMT_kei(x)\n"); return (GMT_d_NaN); } if (x <= 8.0) { /* Telescoped power series from Abramowitz & Stegun */ t = x * 0.125; rxsq = t*t; t = rxsq * rxsq; /* t = pow(x/8, 4) */ return (-log(0.5 * x) * GMT_bei(x) - 0.25 * M_PI * GMT_ber(x) + rxsq * ( 6.76454936 + t * ( -142.91827687 + t * ( 124.23569650 + t * ( -21.30060904 + t * ( 1.17509064 + t * ( -0.02695875 + t * ( 0.00029532 * t)))))))); } else { /* Russell's asymptotic approximation, from Watson, p. 204 */ rxsq = 1.0 / (x * x); t = -x / M_SQRT2; alpha = t; beta = t - 0.125 * M_PI; t *= 0.125 * rxsq; alpha += t; beta -= t; beta -= 0.0625*rxsq; t *= (25.0/48.0)*rxsq; alpha -= t; beta -= t; alpha -= (13.0/128.0)*(rxsq*rxsq); return ( exp(alpha) * sin(beta) / sqrt(2.0 * x / M_PI) ); } } double GMT_i0 (double x) { /* Modified from code in Press et al. */ double y, res; if (x < 0.0) x = -x; if (x < 3.75) { y = x * x / 14.0625; res = 1.0 + y * (3.5156229 + y * (3.0899424 + y * (1.2067492 + y * (0.2659732 + y * (0.360768e-1 + y * 0.45813e-2))))); } else { y = 3.75 / x; res = (exp (x) / sqrt (x)) * (0.39894228 + y * (0.1328592e-1 + y * (0.225319e-2 + y * (-0.157565e-2 + y * (0.916281e-2 + y * (-0.2057706e-1 + y * (0.2635537e-1 + y * (-0.1647633e-1 + y * 0.392377e-2)))))))); } return (res); } double GMT_i1 (double x) { /* Modified Bessel function I1(x) */ double y, res; if (x < 0.0) x = -x; if (x < 3.75) { y = pow (x / 3.75, 2.0); res = x * (0.5 + y * (0.87890594 + y * (0.51498869 + y * (0.15084934 + y * (0.02658733 + y * (0.00301532 + y * 0.00032411)))))); } else { y = 3.75 / x; res = (exp (x) / sqrt (x)) * (0.39894228 + y * (-0.03988024 + y * (-0.00362018 + y * (0.00163801+ y * (-0.01031555 + y * (0.02282967 + y * (-0.02895312 + y * (0.01787654 -y * 0.00420059)))))))); if (x < 0.0) res = - res; } return (res); } double GMT_in (int n, double x) { /* Modified Bessel function In(x) */ int j, m, IACC = 40; double res, tox, bip, bi, bim; double BIGNO = 1.0e10, BIGNI = 1.0e-10; if (n == 0) return (GMT_i0 (x)); if (n == 1) return (GMT_i1 (x)); if (x == 0.0) return (0.0); tox = 2.0 / fabs (x); bip = res = 0.0; bi = 1.0; m = 2 * (n + irint (sqrt (IACC * n))); for (j = m; j >= 1; j--) { bim = bip + ((double)j) * tox * bi; bip = bi; bi = bim; if (fabs (bi) > BIGNO) { res *= BIGNI; bi *= BIGNI; bip *= BIGNI; } if (j == n) res = bip; } res *= (GMT_i0 (x) / bi); if (x < 0.0 && (n%2)) res = -res; return (res); } double GMT_k0 (double x) { /* Modified from code in Press et al. */ double y, z, res; if (x < 0.0) x = -x; if (x <= 2.0) { y = 0.25 * x * x; z = x * x / 14.0625; res = (-log(0.5*x) * (1.0 + z * (3.5156229 + z * (3.0899424 + z * (1.2067492 + z * (0.2659732 + z * (0.360768e-1 + z * 0.45813e-2))))))) + (-M_EULER + y * (0.42278420 + y * (0.23069756 + y * (0.3488590e-1 + y * (0.262698e-2 + y * (0.10750e-3 + y * 0.74e-5)))))); } else { y = 2.0 / x; res = (exp (-x) / sqrt (x)) * (1.25331414 + y * (-0.7832358e-1 + y * (0.2189568e-1 + y * (-0.1062446e-1 + y * (0.587872e-2 + y * (-0.251540e-2 + y * 0.53208e-3)))))); } return (res); } double GMT_k1 (double x) { /* Modified Bessel function K1(x) */ double y, res; if (x < 0.0) x = -x; if (x <= 2.0) { y = x * x / 4.0; res = (log (0.5 * x) * GMT_i1 (x)) + (1.0 / x) * (1.0 + y * (0.15443144 + y * (-0.67278579 + y * (-0.18156897 + y * (-0.01919402 + y * (-0.00110404 - y * 0.00004686)))))); } else { y = 2.0 / x; res = (exp (-x) / sqrt (x)) * (1.25331414 + y * (0.23498619 + y * (-0.03655620 + y * (0.01504268 + y * (-0.00780353 + y * (0.00325614 - y * 0.00068245)))))); } return (res); } double GMT_kn (int n, double x) { /* Modified Bessel function Kn(x) */ int j; double bkm, bk, bkp, tox; if (n == 0) return (GMT_k0 (x)); if (n == 1) return (GMT_k1 (x)); tox = 2.0 / x; bkm = GMT_k0 (x); bk = GMT_k1 (x); for (j = 1; j <= (n-1); j++) { bkp = bkm + j * tox * bk; bkm = bk; bk = bkp; } return (bk); } double GMT_plm (int l, int m, double x) { /* Unnormalized associated Legendre polynomial of degree l and order m, including * Condon-Shortley phase (-1)^m */ double fact, pll = 0, pmm, pmmp1, somx2; int i, ll; /* x is cosine of colatitude and must be -1 <= x <= +1 */ if (fabs(x) > 1.0) { fprintf (stderr, "%s: Error: |x| > 1.0 in GMT_plm_bar\n", GMT_program); return (GMT_d_NaN); } if (m < 0 || m > l) { fprintf (stderr, "%s: Error: GMT_plm requires 0 <= m <= l.\n", GMT_program); return (GMT_d_NaN); } pmm = 1.0; if (m > 0) { somx2 = d_sqrt ((1.0 - x) * (1.0 + x)); fact = 1.0; /* This loop used to go to i < m; corrected to i <= m by WHFS */ for (i = 1; i <= m; i++) { pmm *= -fact * somx2; fact += 2.0; } } if (l == m) return (pmm); pmmp1 = x * (2*m + 1) * pmm; if (l == (m + 1)) return (pmmp1); for (ll = (m+2); ll <= l; ll++) { pll = (x * (2*ll - 1) * pmmp1 - (ll + m - 1) * pmm) / (ll - m); pmm = pmmp1; pmmp1 = pll; } return (pll); } double GMT_plm_bar (int l, int m, double x, BOOLEAN ortho) { /* This function computes the normalized associated Legendre function of x for degree * l and order m. u must be in the range [-1;1] and 0 <= |m| <= l. * The routine is largely based on the second modified forward column method described in * Holmes and Featherstone (2002). It is stable up to very high degree and order * (at least 3000). This is achieved by individually computing the sectorials to P_m,m * and then iterate up to P_l,m divided by P_m,m and the scale factor 1e280. * Eventually, the result is multiplied again with these to terms. * * When ortho=FALSE, normalization is according to the geophysical convention. * - The Condon-Shortley phase (-1)^m is NOT included. * - The normalization factor is sqrt(k*(2l+1)*(l-m)!/(l+m)!) with k=2 except for m=0: k=1. * - The integral of Plm**2 over [-1;1] is 2*k. * - The integrals of (Plm*cos(m*lon))**2 and (Plm*sin(m*lon))**2 over a sphere are 4 pi. * * When ortho=TRUE, the results are orthonormalized. * - The Condon-Shortley phase (-1)^m is NOT included. * - The normalization factor is sqrt((2l+1)*(l-m)!/(l+m)!/(4*pi)). * - The integral of Plm**2 over [-1;1] is 1/(2*pi). * - The integral of (Plm*exp(i*m*lon))**2 over a sphere is 1. * * When called with -m, the Condon-Shortley phase will be included. * * Note that the orthonormalized form produces an integral of 1 when using imaginary terms * for longitude. In geophysics, the imaginary components are split into cosine and sine terms * EACH of which have an average power of 1 over the sphere (i.e. an integral of 4 pi). * * This routine could be further expanded to produce an array of P_m,m through P_l,m. This * would be practical if Legendre functions of various degrees are to be computed at the same * latitude. Also, differentials could be added. * * Reference: * S. A. Holmes and W. E. Featherstone. A unified approach to the Clenshaw summation and the * recursive computation of very high degree and order normalised associated Legendre functions. * Journal of Geodesy, 76, 279-299, 2002. doi:10.1007/s00190-002-0216-2. */ int i; BOOLEAN csphase = FALSE; double scalef=1.0e280, u, r, pmm, pmm0, pmm1, pmm2; /* x is cosine of colatitude (sine of latitude) and must be -1 <= x <= +1 */ if (fabs(x) > 1.0) { fprintf (stderr, "%s: Error: |x| > 1.0 in GMT_plm_bar\n", GMT_program); return (GMT_d_NaN); } /* If m is negative, include Condon-Shortley phase */ if (m < 0) { csphase = TRUE; m = -m; } if (m > l) { fprintf (stderr, "%s: Error: GMT_plm_bar requires 0 <= m <= l.\n", GMT_program); return (GMT_d_NaN); } /* u is sine of colatitude (cosine of latitude) so that 0 <= s <= 1 */ u = d_sqrt ((1.0 - x)*(1.0 + x)); /* Initialize P_00=1 and compute up to P_mm using recurrence relation. The result is a normalisation factor: sqrt((2l+1)(l-m)!/(l+m)!) */ pmm = 1.0; for (i = 1; i <= m; i++) pmm = d_sqrt (1.0 + 0.5/i) * u * pmm; /* If orthonormalization is requested: multiply by sqrt(1/4pi) In case of geophysical convertion : multiply by sqrt(2-delta_0m) */ if (ortho) { pmm *= 0.5 / d_sqrt(M_PI); } else if (m != 0) pmm *= d_sqrt(2.0); /* If C-S phase is requested, apply it now */ if ((m & 1) && csphase) pmm = -pmm; /* If l==m, we are done */ if (l == m) return (pmm); /* In the next section all P_l,m are divided by P_m,m and scaled by factor scalef. First compute P_m+1,m / P_m,m */ pmm0 = 1.0/scalef; pmm1 = pmm0 * x * d_sqrt (2*m + 3); /* Use second modified column forward recurrence relation to compute P_m+2,m / P_m,m */ for (i = m+2; i <= l; i++) { r = (2*i+1.0) / (i+m) / (i-m); pmm2 = x * pmm1 * d_sqrt(r*(2*i-1)) - pmm0 * d_sqrt(r*(i-m-1)*(i+m-1)/(2*i-3)); pmm0 = pmm1; pmm1 = pmm2; } /* Return P_l,m */ pmm1 *= pmm; pmm1 *= scalef; return (pmm1); } /* GMT_sinc (x) calculates the sinc function */ double GMT_sinc (double x) { if (x == 0.0) return (1.0); x *= M_PI; return (sin (x) / x); } /* GMT_factorial (n) calculates the factorial n! */ double GMT_factorial (int n) { int i; double val = 1.0; if (n < 0) { fprintf (stderr, "GMT DOMAIN ERROR: n < 0 in GMT_factorial(n)\n"); return (GMT_d_NaN); /* This could be set to return 0 without warning, to facilitate sums over binomial coefficients, if desired. -whfs */ } for (i = 1; i <= n; i++) val *= ((double)i); return (val); } double GMT_dilog (double x) { /* Compute dilog(x) (defined for x >= 0) by the method of Parker's Appendix A of his Geophysical Inverse Theory. The function is needed for x in the range 0 <= x <= 1 when solving the spherical spline interpolation in section 2.07 of Parker. I tested this for x in range 0 to 1 by reproducting Figure 2.07c of Parker's book. I also tested that the result was smooth for x out to 25. I also checked d[dilog(x)]/dx obtained numerically from this routine against an analytic expression, and I found the smallest angular separation between two points on a sphere such that their Gram matrix expression (see Parker) was different from 1; this is smallest epsilon such that dilog(epsilon) - dilog(0) != 0.0. This turned out to be very much smaller than I would have guessed from the apparent e-15 accuracy of Parker's polynomial. So I think this is very accurate. If this is called with x < 0, it returns dilog(0) and prints a warning. It could also be set to return NaN. I did this because in applications it might happen that an x which should be zero is computed to be slightly off, in which case the desired result is for x = 0. 23 Oct 1996 WHFS. */ double pisqon6, y, ysq, z; if (x < -GMT_CONV_LIMIT) return (GMT_d_NaN); /* Tolerate minor slop before we are outside domain */ pisqon6 = M_PI * M_PI / 6.0; if (x <= 0.0) return (pisqon6); /* Meaning -GMT_CONV_LIMIT < x <= 0 */ if (x < 0.5) { y = -log (1.0 - x); ysq = y * y; z = y * (1.0 + y * (-0.25 + y * (0.027777777777213 + ysq * (-2.7777776990e-04 + ysq * (4.724071696e-06 + ysq * (-9.1764954e-08 + 1.798670e-09 * ysq)))))); return (pisqon6 - z + y * log (x)); } if (x < 2.0) { y = -log (x); ysq = y * y; z = y * (1.0 + y * (-0.25 + y * (0.027777777777213 + ysq * (-2.7777776990e-04 + ysq * (4.724071696e-06 + ysq * (-9.1764954e-08 + 1.798670e-09 * ysq)))))); return (z); } y = log (x); ysq = y * y; z = y * (1.0 + y * (-0.25 + y * (0.027777777777213 + ysq * (-2.7777776990e-04 + ysq * (4.724071696e-06 + ysq * (-9.1764954e-08 + 1.798670e-09 * ysq)))))); return (-z - 0.5 * ysq); } #ifndef M_2_SQRTPI #define M_2_SQRTPI 1.12837916709551257390 #endif double GMT_erfinv (double y) { double x = 0.0, fy, z; /* Misc. efficients for expansion */ static double a[4] = {0.886226899, -1.645349621, 0.914624893, -0.140543331}; static double b[4] = {-2.118377725, 1.442710462, -0.329097515, 0.012229801}; static double c[4] = {-1.970840454, -1.624906493, 3.429567803, 1.641345311}; static double d[2] = {3.543889200, 1.637067800}; fy = fabs (y); if (fy > 1.0) return (GMT_d_NaN); /* Outside domain */ if (GMT_IS_ZERO (1.0 - fy)) { /* Close to +- Inf */ return (copysign (DBL_MAX, y)); } if (y > 0.7) { /* Near upper range */ z = sqrt (-log (0.5 * (1.0 - y))); x = (((c[3] * z + c[2]) * z + c[1]) * z + c[0]) / ((d[1] * z + d[0]) * z + 1.0); } else if (y < -0.7) { /* Near lower range */ z = sqrt (-log (0.5 * (1.0 + y))); x = -(((c[3] * z + c[2]) * z + c[1]) * z + c[0]) / ((d[1] * z + d[0]) * z + 1.0); } else { /* Central range */ z = y * y; x = y * (((a[3] * z + a[2]) * z + a[1]) * z + a[0]) / ((((b[3] * z + b[2]) * z + b[1]) * z + b[0]) * z + 1.0); } /* Apply two steps of Newton-Raphson correction to improve accuracy */ x -= (erf (x) - y) / (M_2_SQRTPI * exp (-x * x)); x -= (erf (x) - y) / (M_2_SQRTPI * exp (-x * x)); return (x); } #if HAVE_ERF == 0 || HAVE_ERFC == 0 /* Need to include the GMT error functions GMT_erf & GMT_erfc * since they are not in the user's local library. */ #ifndef SQRT_PI #define SQRT_PI 0.5641895835477563 #endif static double p[5] = {113.8641541510502, 377.4852376853020, 3209.377589138469, 0.1857777061846032, 3.161123743870566}; static double q[4] = {244.0246379344442, 1282.616526077372, 2844.236833439171, 23.60129095234412}; static double p1[9] = {8.883149794388376, 66.11919063714163, 298.6351381974001, 881.9522212417691, 1712.047612634071, 2051.078377826071, 1230.339354797997, 2.153115354744038e-8, 0.5641884969886701}; static double q1[8] = {117.6939508913125, 537.1811018620099, 1621.389574566690, 3290.799235733460, 4362.619090143247, 3439.367674143722, 1230.339354803749, 15.74492611070983}; static double p2[6] = {-3.603448999498044e-01, -1.257817261112292e-01, -1.608378514874228e-02, -6.587491615298378e-04, -1.631538713730210e-02, -3.053266349612323e-01}; static double q2[5] = {1.872952849923460, 5.279051029514284e-01, 6.051834131244132e-02, 2.335204976268692e-03, 2.568520192289822}; #endif #if HAVE_ERF == 0 double GMT_erf (double y) { int i, sign = 1; double x, res, xsq, xnum, xden, xi; x = y; if (x < 0.0) { sign = -1; x = -x; } if (x < 1.0e-10) { res = x * p[2] / q[2]; } else if (x < 0.477) { xsq = x * x; xnum = p[3] * xsq + p[4]; xden = xsq + q[3]; for (i = 0; i < 3; i++) { xnum = xnum * xsq + p[i]; xden = xden * xsq + q[i]; } res = x * xnum / xden; } else if (x <= 4.0) { xsq = x * x; xnum = p1[7] * x + p1[8]; xden = x + q1[7]; for (i = 0; i < 7; i++) { xnum = xnum * x + p1[i]; xden = xden * x + q1[i]; } res = 1.0 - ((xnum / xden) * exp (-xsq)); } else if (x < 6.375) { xsq = x * x; xi = 1.0 / xsq; xnum = p2[4] * xi + p2[5]; xden = xi + q2[4]; for (i = 0; i < 4; i++) { xnum = xnum * xi + p2[i]; xden = xden * xi + q2[i]; } res = 1.0 - (((SQRT_PI + xi * xnum / xden) / x) * exp (-xsq)); } else res = 1.0; if (sign == -1) res = -res; return (res); } #endif #if HAVE_ERFC == 0 double GMT_erfc (double y) { int i, sign = 1; double x, res, xsq, xnum, xden, xi; x = y; if (x < 0.0) { sign = -1; x = -x; } if (x < 1.0e-10) { res = x * p[2] / q[2]; if (sign == -1) res = -res; res = 1.0 - res; } else if (x < 0.477) { xsq = x * x; xnum = p[3] * xsq + p[4]; xden = xsq + q[3]; for (i = 0; i < 3; i++) { xnum = xnum * xsq + p[i]; xden = xden * xsq + q[i]; } res = x * xnum / xden; if (sign == -1) res = -res; res = 1.0 - res; } else if (x <= 4.0) { xsq = x * x; xnum = p1[7] * x + p1[8]; xden = x + q1[7]; for (i = 0; i < 7; i++) { xnum = xnum * x + p1[i]; xden = xden * x + q1[i]; } res = (xnum / xden) * exp (-xsq); if (sign == -1) res = 2.0 - res; } else if (x < 6.375) { xsq = x * x; xi = 1.0 / xsq; xnum = p2[4] * xi + p2[5]; xden = xi + q2[4]; for (i = 0; i < 4; i++) { xnum = xnum * xi + p2[i]; xden = xden * xi + q2[i]; } res = ((SQRT_PI + xi * xnum / xden) / x) * exp (-xsq); if (sign == -1) res = 2.0 - res; } else res = 0.0; return (res); } #endif #if HAVE_STRDUP == 0 char *GMT_strdup (const char *s) { int n; char *p; n = strlen (s) + 1; p = (char *)malloc ((size_t)n); strncpy (p, s, n); return (p); } #endif #if HAVE_STRTOD == 0 double GMT_strtod (const char *s, char **ends) { /* Given s, try to scan it to convert an ascii string representation of a double, and return the double so found. If (ends != (char **)NULL), return a pointer to the first char of s which cannot be converted. This routine is supplied in GMT because it is not in the POSIX standard. However, it is in ANSI standard C, and so most systems running GMT should have it as a library routine. If the library routine exists, it should be used, as this one will probably be slower. Also, the library routine has ways of dealing with LOCALE info on the radix character, and error setting for over and underflows. Here, I rely on atof() to do that. Note that if s can be converted successfully and a non-null ends was supplied, then on return, *ends[0] == 0. */ char *t, savechar; double x = 0.0; int i, nsign[2], nradix[2], nexp, ndigits, error; BOOLEAN inside = FALSE; t = (char *)s; i = 0; ndigits = 0; while (t[i] && isspace( (int)t[i]) ) i++; if (t[i] == 0 || isalpha ( (int)t[i]) ) { if (ends != (char **)NULL) *ends = (char *)s; return (x); } nsign[0] = nsign[1] = nradix[0] = nradix[1] = nexp = error = 0; while (t[i]) { if (!isdigit((int)t[i])) { switch (t[i]) { case '+': case '-': nsign[nexp]++; if (inside) error++; inside = TRUE; break; case '.': /* This hardwires the radix char, instead of using LOCALE */ nradix[nexp]++; inside = TRUE; break; case 'e': case 'E': nexp++; inside = FALSE; break; default: error++; break; } if (nexp > 1 || nradix[nexp] > 1 || nsign[nexp] > 1) error++; if (error) { if (ndigits == 0) { if (ends != (char **)NULL) *ends = (char *)s; return (0.0); } savechar = t[i]; t[i] = 0; x = atof(t); t[i] = savechar; if (ends != (char **)NULL) *ends = &t[i]; return (x); } } else { ndigits++; inside = TRUE; } i++; } if (ndigits == 0) { if (ends != (char **)NULL) *ends = (char *)s; return (0.0); } x = atof(t); if (ends != (char **)NULL) *ends = &t[i]; return (x); } #endif int GMT_f_q (double chisq1, int nu1, double chisq2, int nu2, double *prob) { /* Routine to compute Q(F, nu1, nu2) = 1 - P(F, nu1, nu2), where nu1 and nu2 are positive integers, chisq1 and chisq2 are random variables having chi-square distributions with nu1 and nu2 degrees of freedom, respectively (chisq1 and chisq2 >= 0.0), F = (chisq1/nu1)/(chisq2/nu2) has the F-distribution, and P(F, nu1, nu2) is the cumulative F-distribution, that is, the integral from 0 to (chisq1/nu1)/(chisq2/nu2) of the F- distribution. Q = 1 - P is small when (chisq1/nu1)/(chisq2/nu2) is large with respect to 1. That is, the value returned by this routine is the likelihood that an F >= (chisq1/nu1)/ (chisq2/nu2) would occur by chance. Follows Abramowitz and Stegun. This is different from the method in Numerical Recipes, which uses the incomplete beta function but makes no use of the fact that nu1 and nu2 are known to be integers, and thus there is a finite limit on the sum for their expression. W H F Smith, August, 1999. REVISED by W H F Smith, October 27, 2000 after GMT 3.3.6 release. I found that the A&S methods overflowed for large nu1 and nu2, so I decided to go back to the GMT_inc_beta way of doing things. */ /* Check range of arguments: */ if (nu1 <= 0 || nu2 <= 0 || chisq1 < 0.0 || chisq2 < 0.0) { fprintf (stderr, "GMT_f_q: Bad argument(s).\n"); return (-1); } /* Extreme cases evaluate immediately: */ if (chisq1 == 0.0) { *prob = 1.0; return (0); } if (chisq2 == 0.0) { *prob = 0.0; return (0); } /* REVISION of Oct 27, 2000: This inc beta call here returns the value. All subsequent code is not used. */ if (GMT_inc_beta(0.5*nu2, 0.5*nu1, chisq2/(chisq2+chisq1), prob) ) { fprintf(stderr,"GMT_q_p: Trouble in GMT_inc_beta call.\n"); return(-1); } return (0); } int GMT_student_t_a(double t, int n, double *prob) { /* Probability integral called A(t,n) by Abramowitz & Stegun for the student's t distribution with n degrees of freedom. Uses expressions A&S 26.7.3 and 26.7.4 If X is distributed N(0,1) and V is distributed chi- square with n degrees of freedom, then tau = X / sqrt(V/n) is said to have Student's t- distribution with n degrees of freedom. For example, tau could be the sample mean divided by the sample standard deviation, for a sample of N points; then n = N - 1. This function sets *prob = GMT_d_NaN and returns (-1) if t < 0. Otherwise it sets *prob = the probability fabs(tau) <= t and returns (0). As n -> oo, we can replace this function with erf (t / M_SQRT2). However, it isn't clear how large n has to be to make this a good approximation. I consulted six books; one of them suggested this approximation for n >= 30, but all the others did not say when to use this approximation (A&S, in particular, does not say). I tried some numerical experiments which suggested that the relative error in this approximation would be < 0.01 for n > 30, all t, but I also found that the expression here is stable to large n and large t, so I decided to leave it as is. W H F Smith, August 1999. */ double theta, s, c, csq, term, sum; int k, kstop, kt, kb; if (t < 0.0 || n <= 0) { fprintf (stderr, "GMT_student_t_a: Bad argument(s).\n"); *prob = GMT_d_NaN; return (-1); } if (t == 0.0) { *prob = 0.0; return (0); } theta = atan(t/sqrt((double)n)); if (n == 1) { *prob = 2.0 * theta / M_PI; return (0); } sincos (theta, &s, &c); csq = c * c; kstop = n-2; if (n%2 == 1) { kt = 0; kb = 1; k = 1; term = c; } else { kt = -1; kb = 0; k = 0; term = 1.0; } sum = term; while (k < kstop) { k += 2; kt += 2; kb += 2; term *= (kt * csq)/kb; sum += term; } sum *= s; if (n%2 == 1) { *prob = 2.0 * (theta + sum) / M_PI; } else { *prob = sum; } /* Adjust in case of roundoff: */ if (*prob < 0.0) *prob = 0.0; if (*prob > 1.0) *prob = 1.0; return (0); } double GMT_zdist (double x) { /* Cumulative Normal (z) distribution */ return (0.5 * (erf (x / M_SQRT2) + 1.0)); } double GMT_zcrit (double alpha) { double sign; /* Critical values for Normal (z) distribution */ /* Simple since z_a = M_SQRT2 * erf^-1 (1-a) */ if (alpha > 0.5) { /* right tail */ alpha = (1.0 - alpha) * 2.0; sign = 1.0; } else { /* left tail */ alpha *= 2.0; sign = -1.0; } return (sign * M_SQRT2 * GMT_erfinv (1.0 - alpha)); } #if 0 /* double qsnorm(p) * double p; * * Function to invert the cumulative normal probability * function. If z is a standardized normal random deviate, * and Q(z) = p is the cumulative Gaussian probability * function, then z = qsnorm(p). * * Note that 0.0 < p < 1.0. Data values outside this range * will return +/- a large number (1.0e6). * To compute p from a sample of data to test for Normalcy, * sort the N samples into non-decreasing order, label them * i=[1, N], and then compute p = i/(N+1). * * Author: Walter H. F. Smith * Date: 19 February, 1991. * * Based on a Fortran subroutine by R. L. Parker. I had been * using IMSL library routine DNORIN(DX) to do what qsnorm(p) * does, when I was at the Lamont-Doherty Geological Observatory * which had a site license for IMSL. I now need to invert the * gaussian CDF without calling IMSL; hence, this routine. * */ double qsnorm (double p) { double t, z; if (p <= 0.0) { fprintf(stderr,"%s: qsnorm: Bad probability.\n", GMT_program); return (-1.0e6); } else if (p >= 1.0) { fprintf(stderr,"%s: qsnorm: Bad probability.\n", GMT_program); return (1.0e6); } else if (p == 0.5) { return (0.0); } else if (p > 0.5) { t = sqrt(-2.0 * log(1.0 - p) ); z = t - (2.515517 +t*(0.802853 +t*0.010328))/ (1.0 + t*(1.432788 + t*(0.189269+ t*0.001308))); return (z); } else { t = sqrt(-2.0 * log(p) ); z = t - (2.515517 +t*(0.802853 +t*0.010328))/ (1.0 + t*(1.432788 + t*(0.189269+ t*0.001308))); return (-z); } } #endif double GMT_tcrit (double alpha, double nu) { /* Critical values for Student t-distribution */ int NU; BOOLEAN done; double t_low, t_high, t_mid, p_high, p_mid, p, sign; if (alpha > 0.5) { /* right tail */ p = 1 - (1 - alpha) * 2.0; sign = 1.0; } else { p = 1 - alpha * 2.0; sign = -1.0; } t_low = GMT_zcrit (alpha); t_high = 5.0; NU = irint(nu); GMT_student_t_a (t_high, NU, &p_high); while (p_high < p) { /* Must pick higher starting point */ t_high *= 2.0; GMT_student_t_a (t_high, NU, &p_high); } /* Now, (t_low, p_low) and (t_high, p_high) are bracketing the desired (t,p) */ done = FALSE; while (!done) { t_mid = 0.5 * (t_low + t_high); GMT_student_t_a (t_mid, NU, &p_mid); if (GMT_IS_ZERO (p_mid - p)) { done = TRUE; } else if (p_mid > p) { /* Too high */ t_high = t_mid; } else { /* p_mid < p */ t_low = t_mid; } } return (sign * t_mid); } double GMT_chi2crit (double alpha, double nu) { /* Critical values for Chi^2-distribution */ BOOLEAN done; double chi2_low, chi2_high, chi2_mid, p_high, p_mid, p; p = 1.0 - alpha; chi2_low = 0.0; chi2_high = 5.0; GMT_chi2 (chi2_high, nu, &p_high); while (p_high > p) { /* Must pick higher starting point */ chi2_high *= 2.0; GMT_chi2 (chi2_high, nu, &p_high); } /* Now, (chi2_low, p_low) and (chi2_high, p_high) are bracketing the desired (chi2,p) */ done = FALSE; while (!done) { chi2_mid = 0.5 * (chi2_low + chi2_high); GMT_chi2 (chi2_mid, nu, &p_mid); if (GMT_IS_ZERO (p_mid - p)) { done = TRUE; } else if (p_mid < p) { /* Too high */ chi2_high = chi2_mid; } else { /* p_mid > p */ chi2_low = chi2_mid; } } return (chi2_mid); } double GMT_Fcrit (double alpha, double nu1, double nu2) { /* Critical values for F-distribution */ int NU1, NU2; BOOLEAN done; double F_low, F_high, F_mid, p_high, p_mid, p, chisq1, chisq2; void F_to_ch1_ch2 (double F, double nu1, double nu2, double *chisq1, double *chisq2); F_high = 5.0; p = 1.0 - alpha; F_low = 0.0; F_to_ch1_ch2 (F_high, nu1, nu2, &chisq1, &chisq2); NU1 = irint (nu1); NU2 = irint (nu2); GMT_f_q (chisq1, NU1, chisq2, NU2, &p_high); while (p_high > p) { /* Must pick higher starting point */ F_high *= 2.0; F_to_ch1_ch2 (F_high, nu1, nu2, &chisq1, &chisq2); GMT_f_q (chisq1, NU1, chisq2, NU2, &p_high); } /* Now, (F_low, p_low) and (F_high, p_high) are bracketing the desired (F,p) */ done = FALSE; while (!done) { F_mid = 0.5 * (F_low + F_high); F_to_ch1_ch2 (F_mid, nu1, nu2, &chisq1, &chisq2); GMT_f_q (chisq1, NU1, chisq2, NU2, &p_mid); if (GMT_IS_ZERO (p_mid - p)) { done = TRUE; } else if (p_mid < p) { /* Too high */ F_high = F_mid; } else { /* p_mid > p */ F_low = F_mid; } } return (F_mid); } void F_to_ch1_ch2 (double F, double nu1, double nu2, double *chisq1, double *chisq2) { /* Silly routine to break F up into parts needed for GMT_f_q */ *chisq2 = 1.0; *chisq1 = F * nu1 / nu2; } #if HAVE_J0 == 0 /* Alternative j0 coded from Numerical Recipes by Press et al */ double GMT_j0 (double x) { double ax, z, xx, y, ans, ans1, ans2, s, c; if ((ax = fabs (x)) < 8.0) { y = x * x; ans1 = 57568490574.0 + y * (-13362590354.0 + y * (651619640.7 + y * (-11214424.18 + y * (77392.33017 + y * (-184.9052456))))); ans2 = 57568490411.0 + y * ( 1029532985.0 + y * ( 9494680.718 + y * ( 59272.64853 + y * ( 267.8532712 + y * 1.0)))); ans = ans1 / ans2; } else { z = 8.0 / ax; y = z * z; xx = ax - 0.785398164; ans1 = 1.0 + y * (-0.1098628627e-2 + y * ( 0.2734510407e-4 + y * (-0.2073370639e-5 + y * (0.2093887211e-6)))); ans2 = -0.1562499995e-1 + y * ( 0.1430488765e-3 + y * (-0.6911147651e-5 + y * ( 0.7621095161e-6 + y * (-0.934935152e-7)))); sincos (xx, &s, &c); ans = d_sqrt (0.636619772 / ax) * (c * ans1 - z * s * ans2); } return (ans); } #endif #if HAVE_J1 == 0 /* Alternative j1 coded from Numerical Recipes by Press et al */ double GMT_j1 (double x) { double ax, z, xx, y, ans, ans1, ans2, s, c; if ((ax = fabs (x)) < 8.0) { y = x * x; ans1 = x * (72362614232.0 + y * (-7895059235.0 + y * (242396853.1 + y * (-2972611.439 + y * (15704.48260 + y * (-30.16036606)))))); ans2 = 144725228442.0 + y * ( 2300535178.0 + y * ( 18583304.74 + y * ( 99447.43394 + y * ( 376.9991397 + y * 1.0)))); ans = ans1 / ans2; } else { z = 8.0 / ax; y = z * z; xx = ax - 2.356194491; ans1 = 1.0 + y * ( 0.183105e-2 + y * (-0.3516396496e-4 + y * ( 0.2457520174e-5 + y * (-0.240337019e-6)))); ans2 = 0.04687499995 + y * ( -0.2002690873e-3 + y * ( 0.8449199096e-5 + y * (-0.88228987e-6 + y * ( 0.105787412e-6)))); sincos (xx, &s, &c); ans = d_sqrt (0.636619772 / ax) * (c * ans1 - z * s * ans2); } return (ans); } #endif #if HAVE_JN == 0 #define ACC 40.0 #define BIGNO 1.0e10 #define BIGNI 1.0e-10 /* Alternative jn coded from Numerical Recipes by Press et al */ double GMT_jn (int n, double x) { int j, jsum, m; double ax, bj, bjm, bjp, sum, tox, ans; if (n == 0) return (GMT_j0 (x)); if (n == 1) return (GMT_j1 (x)); ax = fabs (x); if (GMT_IS_ZERO (ax)) return (0.0); if (ax > (double)n) { /* Upwards recurrence */ tox = 2.0 / ax; bjm = GMT_j0 (ax); bj = GMT_j1 (ax); for (j = 1; j < n; j++) { bjp = (double)j * tox * bj - bjm; bjm = bj; bj = bjp; } ans = bj; } else { /* More complicated here */ tox = 2.0 / ax; m = 2 * ((n + (int) d_sqrt(ACC * n)) / 2); jsum = 0; bjp = ans = sum = 0.0; bj = 1.0; for (j = m; j > 0; j--) { bjm = (double)j * tox * bj - bjp; bjp = bj; bj = bjm; if (fabs (bj) > BIGNO) { bj *= BIGNI; bjp *= BIGNI; ans *= BIGNI; sum *= BIGNI; } if (jsum) sum += bj; jsum = !jsum; if (j == n) ans = bjp; } sum = 2.0 * sum - bj; ans /= sum; } return ((x < 0.0 && (n % 2) == 1) ? -ans : ans); } #endif #if HAVE_Y0 == 0 /* Alternative y0, y1, yn coded from Numerical Recipes by Press et al */ double GMT_y0 (double x) { double z, ax, xx, y, ans, ans1, ans2, s, c; if (x < 8.0) { y = x * x; ans1 = -2957821389.0 + y * (7062834065.0 + y * (-512359803.6 + y * (10879881.29 + y * (-86327.92757 + y * (228.4622733))))); ans2 = 40076544269.0 + y * ( 745249964.8 + y * ( 7189466.438 + y * ( 47447.26470 + y * ( 226.1030244 + y * 1.0)))); ans = (ans1 / ans2) + 0.636619772 * GMT_j0 (x) * d_log (x); } else { z = 8.0 / x; ax = fabs (x); y = z * z; xx = x - 0.785398164; ans1 = 1.0 + y * (-0.1098628627e-2 + y * ( 0.2734510407e-4 + y * (-0.2073370639e-5 + y * (0.2093887211e-6)))); ans2 = -0.1562499995e-1 + y * ( 0.1430488765e-3 + y * (-0.6911147651e-5 + y * ( 0.7621095161e-6 + y * (-0.934935152e-7)))); sincos (xx, &s, &c); ans = d_sqrt (0.636619772 / ax) * (s * ans1 - z * c * ans2); } fprintf (stderr, "GMT_y0 called\n"); return (ans); } #endif #if HAVE_Y1 == 0 /* Alternative y1 coded from Numerical Recipes by Press et al */ double GMT_y1 (double x) { double z, ax, xx, y, ans, ans1, ans2, s, c; if (x < 8.0) { y = x * x; ans1 = x * (-0.4900604943e13 + y * (0.1275274390e13 + y * (-0.5153438139e11 + y * (0.7349264551e9 + y * (-0.4237922726e7 + y * (0.8511937935e4)))))); ans2 = 0.2499580570e14 + y * (0.4244419664e12 + y * ( 0.3733650367e10 + y * (0.2245904002e8 + y * ( 0.1020426050e6 + y * (0.3549632885e3) + y)))); ans = (ans1 / ans2) + 0.636619772 * (GMT_j1 (x) * d_log (x) - 1.0 / x); } else { z = 8.0 / x; ax = fabs (x); y = z * z; xx = x - 2.356194491; ans1 = 1.0 + y * (0.183105e-2 + y * ( -0.3516396496e-4 + y * (0.2457520174e-5 + y * (-0.240337019e-6)))); ans2 = 0.04687499995 + y * ( -0.2002690873e-3 + y * (0.8449199096e-5 + y * ( -0.88228987e-6 + y * (0.105787412e-6)))); sincos (xx, &s, &c); ans = d_sqrt (0.636619772 / ax) * (s * ans1 - z * c * ans2); } return (ans); } #endif #if HAVE_YN == 0 /* Alternative yn coded from Numerical Recipes by Press et al */ double GMT_yn (int n, double x) { int j; double by, bym, byp, tox; if (n == 0) return (GMT_y0 (x)); if (n == 1) return (GMT_y1 (x)); tox = 2.0 / x; by = GMT_y1 (x); bym = GMT_y0 (x); for (j = 1; j < n; j++) { byp = (double)j * tox * by - bym; bym = by; by = byp; } return (by); } #endif #if HAVE_SINCOS == 0 && HAVE_ALPHASINCOS == 0 /* Platform does not have sincos - make a dummy one with sin and cos */ void sincos (double a, double *s, double *c) { *s = sin (a); *c = cos (a); } #endif #define GMT_RAND_IQ 127773 #define GMT_RAND_IA 16807 #define GMT_RAND_IM INT_MAX #define GMT_RAND_AM (1.0 / GMT_RAND_IM) #define GMT_RAND_IR 2836 #define GMT_RAND_NTAB 32 #define GMT_RAND_NDIV (1 + (GMT_RAND_IM - 1) / GMT_RAND_NTAB) #define GMT_RAND_EPS 2.2204460492503131E-16 double GMT_rand () { /* Uniform random number generator based on ran1 of * Press et al, Numerical Recipes, 2nd edition, * converted from Fortran to C. Will return values * x so that 0.0 < x < 1.0 occurs with equal probability. */ static int GMT_rand_iy = 0; static int GMT_rand_seed = 0; static int GMT_rand_iv[GMT_RAND_NTAB]; int j, k; if (GMT_rand_iy == 0) { /* First time initialization */ GMT_rand_seed = (int) time (NULL); /* Seed value is positive sec since 1970 */ if (GMT_rand_seed <= 0) GMT_rand_seed = 1; /* Make sure to prevent seed = 0 */ for (j = GMT_RAND_NTAB + 8 - 1; j >= 0; j--) { /* Load shuffle table after 8 warm-ups */ k = GMT_rand_seed / GMT_RAND_IQ; GMT_rand_seed = GMT_RAND_IA * (GMT_rand_seed - k * GMT_RAND_IQ) - GMT_RAND_IR * k; if (GMT_rand_seed < 0) GMT_rand_seed += GMT_RAND_IM; if (j < GMT_RAND_NTAB) GMT_rand_iv[j] = GMT_rand_seed; } GMT_rand_iy = GMT_rand_iv[0]; } /* Starts here when not initializing */ k = GMT_rand_seed / GMT_RAND_IQ; /* Compute GMT_rand_seed = mod (GMT_rand_seed * GMT_RAND_IA, GMT_RAND_IM) without overflowing using Schrage's method */ GMT_rand_seed = GMT_RAND_IA * (GMT_rand_seed - k * GMT_RAND_IQ) - GMT_RAND_IR * k; if (GMT_rand_seed < 0) GMT_rand_seed += GMT_RAND_IM; j = GMT_rand_iy / GMT_RAND_NDIV; /* Will be in the range 0:NTAB-1 */ GMT_rand_iy = GMT_rand_iv[j]; /* Output previous shuffle value and fill table again */ GMT_rand_iv[j] = GMT_rand_seed; return (MIN (GMT_RAND_AM * GMT_rand_iy, (1.0 - GMT_RAND_EPS))); /* so we don't return 1.0 */ } double GMT_nrand (void) { /* Gaussian random number generator based on gasdev of * Press et al, Numerical Recipes, 2nd edition. Will * return values that have zero mean and unit variance. */ static int iset = 0; static double gset; double fac, r, v1, v2; if (iset == 0) { /* We don't have an extra deviate handy, so */ do { v1 = 2.0 * GMT_rand () - 1.0; /* Pick two uniform numbers in the -1/1/-1/1 square */ v2 = 2.0 * GMT_rand () - 1.0; r = v1 * v1 + v2 * v2; } while (r >= 1.0 || r == 0.0); /* Keep trying until v1,v2 is inside unit circle */ fac = sqrt (-2.0 * log (r) / r); /* Now make Box-Muller transformation to get two normal deviates. Return * one and save the other for the next time GMT_nrand is called */ gset = v1 * fac; iset = 1; /* Set flag for next time */ return (v2 * fac); } else { iset = 0; /* Take old value, reset flag */ return (gset); } } double GMT_lrand (void) { /* Laplace random number generator. As nrand, it will * return values that have zero mean and unit variance. */ double rand_0_to_1; rand_0_to_1 = GMT_rand (); /* Gives uniformly distributed random values in 0-1 range */ return ( ((rand_0_to_1 <= 0.5) ? log (2.0 * rand_0_to_1) : -log (2.0 * (1.0 - rand_0_to_1))) / M_SQRT2); } void GMT_chi2 (double chi2, double nu, double *prob) { /* Evaluate probability that chi2 will exceed the * theoretical chi2 by chance. */ *prob = GMT_gammq (0.5 * nu, 0.5 * chi2); } void GMT_cumpoisson (double k, double mu, double *prob) { /* evaluate Cumulative Poisson Distribution */ *prob = (k == 0.0) ? exp (-mu) : GMT_gammq (k, mu); } #if HAVE_HYPOT == 0 double GMT_hypot (double x, double y) { /* Return sqrt(x*x + y*y), guarding against some overflows where possible. If a local library hypot() is available, it should be used instead; it could be faster and more accurate. WHFS 30 March 2000 */ double a, b, c, d, r, s, t; if (GMT_is_dnan (x) || GMT_is_dnan (y) ) return (GMT_d_NaN); /* A complete implementation of IEEE exceptional values would also return +Inf if either x or y is +/- Inf */ if (x == 0.0) return (fabs(y)); if (y == 0.0) return (fabs(x)); a = fabs(x); b = fabs(y); /* If POSIX defined M_SQRT2 (= sqrt(2.0) to machine precision) then we could say if (a == b) return (a * M_SQRT2); */ if (a < b) { c = b; d = a/b; } else if (a == b) { return (a * M_SQRT2); /* Added 17 Aug 2001 */ } else { c = a; d = b/a; } /* DBL_EPSILON is defined in POSIX float.h as the smallest positive e such that 1+e > 1 in floating point. */ if (d < DBL_EPSILON) return (c); s = d*d; if (s > DBL_EPSILON) return (c * sqrt (1.0 + s) ); t = 1.0 + d; s = (2 * (d/t)) / t; r = t * sqrt (1.0 - s); if (r <= 1.0) return (c); return (c * r); } #endif #if HAVE_LOG1P == 0 double GMT_log1p (double x) { /* Approximate log(1 + x) fairly well for small x. This should be used only if there isn't a better library routine available. WHFS 30 March 2000 */ double u; if (GMT_is_dnan(x) || x <= -1.0) return (GMT_d_NaN); u = 1.0 + x; if (u == 1.0) return (x); else return ( log(u) * (x/(u - 1.0) ) ); } #endif #if HAVE_ATANH == 0 double GMT_atanh (double x) { /* Return hyperbolic arctangent of x. This should be used only if there isn't a better library routine available. WHFS 30 March 2000 */ if (GMT_is_dnan(x) || fabs(x) >= 1.0) return (GMT_d_NaN); return (0.5 * ( log1p(x) - log1p(-x) ) ); } #endif int GMT_median (double *x, size_t n, double xmin, double xmax, double m_initial, double *med) { double lower_bound, upper_bound, m_guess, t_0, t_1, t_middle; double lub, glb, xx, temp; size_t i, n_above, n_below, n_equal, n_lub, n_glb; int finished = FALSE; int iteration = 0; if (n == 0) { *med = m_initial; return (1); } if (n == 1) { *med = x[0]; return (1); } if (n == 2) { *med = 0.5 * (x[0] + x[1]); return (1); } m_guess = m_initial; lower_bound = xmin; upper_bound = xmax; t_0 = 0; t_1 = n - 1; t_middle = 0.5 * (n - 1); do { n_above = n_below = n_equal = n_lub = n_glb = 0; lub = xmax; glb = xmin; for (i = 0; i < n; i++) { xx = x[i]; if (xx == m_guess) { n_equal++; } else if (xx > m_guess) { n_above++; if (xx < lub) { lub = xx; n_lub = 1; } else if (xx == lub) { n_lub++; } } else { n_below++; if (xx > glb) { glb = xx; n_glb = 1; } else if (xx == glb) { n_glb++; } } } iteration++; /* Now test counts, watch multiple roots, think even/odd: */ if ((labs((long)(n_above - n_below))) <= (long)n_equal) { if (n_equal) { *med = m_guess; } else { *med = 0.5 * (lub + glb); } finished = TRUE; } else if ((labs ((long)((n_above - n_lub) - (n_below + n_equal)))) < (long)n_lub) { *med = lub; finished = TRUE; } else if ((labs ((long)((n_below - n_glb) - (n_above + n_equal)))) < (long)n_glb) { *med = glb; finished = TRUE; } /* Those cases found the median; the next two will forecast a new guess: */ else if ( n_above > (n_below + n_equal) ) { /* Guess is too low */ lower_bound = m_guess; t_0 = n_below + n_equal - 1; temp = lower_bound + (upper_bound - lower_bound) * (t_middle - t_0) / (t_1 - t_0); m_guess = (temp > lub) ? temp : lub; /* Move guess at least to lub */ } else if ( n_below > (n_above + n_equal) ) { /* Guess is too high */ upper_bound = m_guess; t_1 = n_below + n_equal - 1; temp = lower_bound + (upper_bound - lower_bound) * (t_middle - t_0) / (t_1 - t_0); m_guess = (temp < glb) ? temp : glb; /* Move guess at least to glb */ } else { /* If we get here, I made a mistake! */ fprintf (stderr,"%s: GMT Fatal Error: Internal goof - please report to developers!\n", GMT_program); GMT_exit (EXIT_FAILURE); } } while (!finished); /* That's all, folks! */ return (iteration); } int GMT_mode (double *x, size_t n, size_t j, int sort, int mode_selection, int *n_multiples, double *mode_est) { size_t i, istop; int multiplicity; double mid_point_sum = 0.0, length, short_length = DBL_MAX, this_mode; if (n == 0) return (0); if (n == 1) { *mode_est = x[0]; return (0); } if (sort) qsort((void *)x, (size_t)n, sizeof(double), GMT_comp_double_asc); istop = n - j; multiplicity = 0; for (i = 0; i < istop; i++) { length = x[i + j] - x[i]; if (length < 0.0) { fprintf(stderr,"GMT_mode: Array not sorted in non-decreasing order.\n"); return (-1); } else if (length == short_length) { /* Possibly multiple mode */ switch (mode_selection) { case -1: /* Always pick lowest mode */ this_mode = 0.5 * (x[i + j] + x[i]); if (this_mode < mid_point_sum) mid_point_sum = this_mode; break; case 0: /* Return average of all modes */ multiplicity++; mid_point_sum += (0.5 * (x[i + j] + x[i])); break; case +1: /* Always pick highest mode */ this_mode = 0.5 * (x[i + j] + x[i]); if (this_mode > mid_point_sum) mid_point_sum = this_mode; break; } } else if (length < short_length) { /* Update current best mode estimate */ multiplicity = 1; mid_point_sum = (0.5 * (x[i + j] + x[i])); short_length = length; } } if (multiplicity > 1) { /* Found more than 1 mode; return mean of them all */ mid_point_sum /= multiplicity; (*n_multiples) += multiplicity; } *mode_est = mid_point_sum; return (0); } int GMT_mode_f (float *x, size_t n, size_t j, int sort, int mode_selection, int *n_multiples, double *mode_est) { size_t i, istop; int multiplicity; double mid_point_sum = 0.0, length, short_length = FLT_MAX, this_mode; if (n == 0) return (0); if (n == 1) { *mode_est = x[0]; return (0); } if (sort) qsort((void *)x, (size_t)n, sizeof(float), GMT_comp_float_asc); istop = n - j; multiplicity = 0; for (i = 0; i < istop; i++) { length = x[i + j] - x[i]; if (length < 0.0) { fprintf (stderr,"GMT_mode_f: Array not sorted in non-decreasing order.\n"); return (-1); } else if (length == short_length) { /* Possibly multiple mode */ switch (mode_selection) { case -1: /* Always pick lowest mode */ this_mode = 0.5 * (x[i + j] + x[i]); if (this_mode < mid_point_sum) mid_point_sum = this_mode; break; case 0: /* Return average of all modes */ multiplicity++; mid_point_sum += (0.5 * (x[i + j] + x[i])); break; case +1: /* Always pick highest mode */ this_mode = 0.5 * (x[i + j] + x[i]); if (this_mode > mid_point_sum) mid_point_sum = this_mode; break; } } else if (length < short_length) { multiplicity = 1; mid_point_sum = (0.5 * (x[i + j] + x[i])); short_length = length; } } if (multiplicity > 1) { /* Found more than 1 mode; return mean of them all */ mid_point_sum /= multiplicity; (*n_multiples) += multiplicity; } *mode_est = mid_point_sum; return (0); } /* Replacement slower functions until we figure out the problem with the algorithm */ void GMT_getmad (double *x, size_t n, double location, double *scale) { size_t i; double *dev, med; dev = (double *) GMT_memory (VNULL, n, sizeof(double), GMT_program); for (i = 0; i < n; i++) dev[i] = fabs (x[i] - location); qsort ((void *)dev, (size_t)n, sizeof (double), GMT_comp_double_asc); for (i = n; GMT_is_dnan (dev[i-1]) && i > 1; i--); if (i) med = (i%2) ? dev[i/2] : 0.5 * (dev[(i-1)/2] + dev[i/2]); else med = GMT_d_NaN; GMT_free ((void *)dev); *scale = 1.4826 * med; } void GMT_getmad_f (float *x, size_t n, double location, double *scale) { size_t i; float *dev; double med; dev = (float *) GMT_memory (VNULL, n, sizeof(float), GMT_program); for (i = 0; i < n; i++) dev[i] = (float) fabs ((double)(x[i] - location)); qsort ((void *)dev, (size_t)n, sizeof (float), GMT_comp_float_asc); for (i = n; GMT_is_fnan (dev[i-1]) && i > 1; i--); if (i) med = (i%2) ? dev[i/2] : 0.5 * (dev[(i-1)/2] + dev[i/2]); else med = GMT_d_NaN; GMT_free ((void *)dev); *scale = 1.4826 * med; } void GMT_getmad_BROKEN (double *x, int n, double location, double *scale) { /* Compute MAD (Median Absolute Deviation) for a double data set */ double e_low, e_high, error, last_error; int i_low, i_high, n_dev, n_dev_stop; i_low = 0; while (i_low < n && x[i_low] <= location) i_low++; i_low--; i_high = n - 1; while (i_high >= 0 && x[i_high] >= location) i_high--; i_high++; while (i_high < i_low) i_high++, i_low--; /* I think this must be added in (P. Wessel, 9/29/04) */ n_dev_stop = n / 2; error = last_error = 0.0; n_dev = 0; while (n_dev < n_dev_stop) { last_error = error; if (i_low < 0) { error = x[i_high] - location; i_high++; n_dev++; } else if (i_high == n) { error = location - x[i_low]; i_low--; n_dev++; } else { e_low = location - x[i_low]; e_high = x[i_high] - location; if (e_low < e_high) { error = e_low; i_low--; n_dev++; } else if (e_high < e_low) { error = e_high; i_high++; n_dev++; } else { error = e_high; i_low--; i_high++; if ( !(n_dev)) n_dev--; /* First time count only once */ n_dev += 2; } } } *scale = (n%2) ? (1.4826 * error) : (0.7413 * (error + last_error)); } void GMT_getmad_f_BROKEN (float *x, int n, double location, double *scale) { /* Compute MAD (Median Absolute Deviation) for a float data set */ double e_low, e_high, error, last_error; int i_low, i_high, n_dev, n_dev_stop; i_low = 0; while (i_low < n && x[i_low] <= location) i_low++; i_low--; i_high = n - 1; while (i_high >= 0 && x[i_high] >= location) i_high--; i_high++; while (i_high < i_low) i_high++, i_low--; /* I think this must be added in (P. Wessel, 9/29/04) */ n_dev_stop = n / 2; error = last_error = 0.0; n_dev = 0; while (n_dev < n_dev_stop) { last_error = error; if (i_low < 0) { error = x[i_high] - location; i_high++; n_dev++; } else if (i_high == n) { error = location - x[i_low]; i_low--; n_dev++; } else { e_low = location - x[i_low]; e_high = x[i_high] - location; if (e_low < e_high) { error = e_low; i_low--; n_dev++; } else if (e_high < e_low) { error = e_high; i_high++; n_dev++; } else { error = e_high; i_low--; i_high++; if ( !(n_dev)) n_dev--; /* First time count only once */ n_dev += 2; } } } *scale = (n%2) ? (1.4826 * error) : (0.7413 * (error + last_error)); } double GMT_extreme (double x[], size_t n, double x_default, int kind, int way) { /* Returns the extreme value in the x array according to: * kind: -1 means only consider negative values. * kind: 0 means consider all values. * kind: +1 means only consider positive values. * way: -1 means look for mimimum. * way: +1 means look for maximum. * If kind is non-zero we assign x_default is no values are found. * If kind == 0 we expect x_default to be set so that x[0] will reset x_select. */ size_t i, k; double x_select = GMT_d_NaN; for (i = k = 0; i < n; i++) { if (kind == -1 && x[i] > 0.0) continue; if (kind == +1 && x[i] < 0.0) continue; if (k == 0) x_select = x[i]; if (way == -1 && x[i] < x_select) x_select = x[i]; if (way == +1 && x[i] > x_select) x_select = x[i]; k++; } return ((k) ? x_select : x_default); } int GMT_chebyshev (double x, int n, double *t) { /* Calculates the n'th Chebyshev polynomial at x */ double x2, a, b; if (n < 0) GMT_err_pass (GMT_CHEBYSHEV_NEG_ORDER, ""); if (fabs (x) > 1.0) GMT_err_pass (GMT_CHEBYSHEV_BAD_DOMAIN, ""); switch (n) { /* Testing the order of the polynomial */ case 0: *t = 1.0; break; case 1: *t = x; break; case 2: *t = 2.0 * x * x - 1.0; break; case 3: *t = x * (4.0 * x * x - 3.0); break; case 4: x2 = x * x; *t = 8.0 * x2 * (x2 - 1.0) + 1.0; break; default: /* For higher degrees we do the recursion */ GMT_chebyshev (x, n-1, &a); GMT_chebyshev (x, n-2, &b); *t = 2.0 * x * a - b; break; } return (GMT_NOERROR); } double GMT_corrcoeff (double *x, double *y, size_t n, int mode) { /* Returns plain correlation coefficient, r. * If mode = 1 we assume mean(x) = mean(y) = 0. */ size_t i, n_use; double xmean = 0.0, ymean = 0.0, dx, dy, vx, vy, vxy, r; if (mode == 0) { for (i = n_use = 0; i < n; i++) { if (GMT_is_dnan (x[i]) || GMT_is_dnan (y[i])) continue; xmean += x[i]; ymean += y[i]; n_use++; } if (n_use == 0) return (GMT_d_NaN); xmean /= (double)n_use; ymean /= (double)n_use; } vx = vy = vxy = 0.0; for (i = n_use = 0; i < n; i++) { if (GMT_is_dnan (x[i]) || GMT_is_dnan (y[i])) continue; dx = x[i] - xmean; dy = y[i] - ymean; vx += dx * dx; vy += dy * dy; vxy += dx * dy; } r = vxy / sqrt (vx * vy); return (r); } double GMT_corrcoeff_f (float *x, float *y, size_t n, int mode) { /* Returns plain correlation coefficient, r. * If mode = 1 we assume mean(x) = mean(y) = 0. */ size_t i, n_use; double xmean = 0.0, ymean = 0.0, dx, dy, vx, vy, vxy, r; if (mode == 0) { for (i = n_use = 0; i < n; i++) { if (GMT_is_fnan (x[i]) || GMT_is_fnan (y[i])) continue; xmean += (double)x[i]; ymean += (double)y[i]; n_use++; } if (n_use == 0) return (GMT_d_NaN); xmean /= (double)n_use; ymean /= (double)n_use; } vx = vy = vxy = 0.0; for (i = n_use = 0; i < n; i++) { if (GMT_is_fnan (x[i]) || GMT_is_fnan (y[i])) continue; dx = (double)x[i] - xmean; dy = (double)y[i] - ymean; vx += dx * dx; vy += dy * dy; vxy += dx * dy; } r = vxy / sqrt (vx * vy); return (r); } double GMT_quantile (double *x, double q, size_t n) { /* Returns the q'th (q in percent) quantile of x (assumed sorted). * q is expected to be 0 < q < 100 */ int i_f; double p, f, df; while (n > 1 && GMT_is_dnan (x[n-1])) n--; /* Skip any NaNs at the end of x */ if (n < 1) return (GMT_d_NaN); /* Need at least 1 point to do something */ if (q == 0.0) return (x[0]); /* 0% quantile == min(x) */ if (q == 100.0) return (x[n-1]); /* 100% quantile == max(x) */ f = (n - 1) * q / 100.0; i_f = (int)floor (f); if ((df = (f - (double)i_f)) > 0.0) /* Must interpolate between the two neighbors */ p = x[i_f+1] * df + x[i_f] * (1.0 - df); else /* Exactly on a node */ p = x[i_f]; return (p); } double GMT_quantile_f (float *x, double q, size_t n) { /* Returns the q'th (q in percent) quantile of x (assumed sorted). * q is expected to be 0 < q < 100 */ int i_f; double p, f, df; while (n > 1 && GMT_is_fnan (x[n-1])) n--; /* Skip any NaNs at the end of x */ if (n < 1) return (GMT_d_NaN); /* Need at least 1 point to do something */ if (q == 0.0) return ((double)x[0]); /* 0% quantile == min(x) */ if (q == 100.0) return ((double)x[n-1]); /* 100% quantile == max(x) */ f = (n - 1) * q / 100.0; i_f = (int)floor (f); if ((df = (f - (double)i_f)) > 0.0) /* Must interpolate between the two neighbors */ p = (double)(x[i_f+1] * df + x[i_f] * (1.0 - df)); else /* Exactly on a node */ p = (double)x[i_f]; return (p); } static void Cmul (double A[], double B[], double C[]) { /* Complex multiplication */ C[0] = A[0]*B[0] - A[1]*B[1]; C[1] = A[0]*B[1] + A[1]*B[0]; } static void Cdiv (double A[], double B[], double C[]) { /* Complex division */ double denom; denom = B[0]*B[0] + B[1]*B[1]; C[0] = (A[0]*B[0] + A[1]*B[1])/denom; C[1] = (A[1]*B[0] - A[0]*B[1])/denom; } double GMT_psi (double zz[], double p[]) { /* Psi Psi (or Digamma) function for complex arguments z. * * d * Psi(z) = --log(Gamma(z)) * dz * * zz[0] is real and zz[1] is imaginary component; same for p on output (only if p != NULL). * We also return the real component as function result. */ static double c[15] = { 0.99999999999999709182, 57.156235665862923517, -59.597960355475491248, 14.136097974741747174, -0.49191381609762019978, 0.33994649984811888699e-4, 0.46523628927048575665e-4, -0.98374475304879564677e-4, 0.15808870322491248884e-3, -0.21026444172410488319e-3, 0.21743961811521264320e-3, -0.16431810653676389022e-3, 0.84418223983852743293e-4, -0.26190838401581408670e-4, 0.36899182659531622704e-5}; double z[2], g[2], dx[2], dd[2], d[2], n[2], gg[2], f[2], x0, A[2], B[2], C[2], sx, cx, e; int k; if (zz[1] == 0.0 && rint(zz[0]) == zz[0] && zz[0] <= 0.0) { if (p) { p[0] = GMT_d_NaN; p[1] = 0.0;} return (GMT_d_NaN); /* Singular points */ } z[0] = zz[0]; z[1] = zz[1]; if ((x0 = z[0]) < 0.5) { /* reflection point */ z[0] = 1.0 - z[0]; z[1] = -z[1]; } /* Lanczos approximation */ g[0] = 607.0/128.0; g[1] = 0.0; /* best results when 4<=g<=5 */ n[0] = d[0] = n[1] = d[1] = 0.0; for (k = 14; k > 0; k--) { A[0] = 1.0; A[1] = 0.0; B[0] = z[0] + k - 1.0; B[1] = z[1]; Cdiv (A, B, dx); dd[0] = c[k] * dx[0]; dd[1] = c[k] * dx[1]; d[0] += dd[0]; d[1] += dd[1]; Cmul (dd, dx, B); n[0] -= B[0]; n[1] -= B[1]; } d[0] += c[0]; gg[0] = z[0] + g[0] - 0.5; gg[1] = z[1]; Cdiv (n, d, A); Cdiv (g, gg, B); f[0] = log (hypot(gg[0], gg[1])) + A[0] - B[0]; f[1] = atan2 (gg[1], gg[0]) + A[1] - B[1]; if (x0 < 0.5) { C[0] = M_PI * zz[0]; C[1] = M_PI * zz[1]; e = exp (-2*C[1]); sx = sin (2*C[0]); cx = cos (2*C[0]); A[0] = -e * sx; A[1] = e * cx + 1.0; B[0] = e * cx - 1.0; B[1] = e * sx; Cdiv (A, B, C); f[0] -= M_PI * C[0]; f[1] -= M_PI * C[1]; } if (p) { p[0] = f[0]; p[1] = f[1]; } return (f[0]); } #ifndef M_SQRT_PI #define M_SQRT_PI 1.772453850905516 #endif void GMT_PvQv (double x, double v_ri[], double pq[], int *iter) { /* Here, -1 <= x <= +1, v_ri is an imaginary number [r,i], and we return * both Pv(x) and Qv(x) in the pq array since no savings in doing it separately. * Based on recipe in Spanier/Oldham: An Atlas of Functions. * pq[0-1] is real/imag Pv * pq[2-3] is real/imag Qv */ double v[2], a[2], R[2], r[2], z[2], tmp[2], X[2], ep, em, s[2], c[2]; double M, L, K, vp1[2], G[2], Xn, x2, g, u, t, f, k, k1, sx, cx, w, sL, cL; *iter = 0; if (x == -1) { /* Both blow up */ pq[0] = pq[2] = GMT_d_NaN; pq[1] = pq[3] = 0.0; return; } else if (x == +1) { /* Pv = 1, Qv = inf */ pq[0] = 1; pq[2] = GMT_d_NaN; pq[1] = pq[3] = 0.0; return; } /* General case of |x| < 1 */ a[0] = a[1] = R[0] = 1.0; R[1] = 0.0; v[0] = v_ri[0]; v[1] = v_ri[1]; Cmul (v, v, z); z[0] = v[0] - z[0]; z[1] = v[1] - z[1]; K = 4.0 * sqrt (hypot(z[0], z[1])); vp1[0] = v[0] + 1.0; vp1[1] = v[1]; if ((hypot(vp1[0], vp1[1]) + floor(vp1[1])) == 0.0) { a[0] = GMT_d_NaN; a[1] = 0.0; v[0] = -1.0 - v[0]; v[1] = -v[1]; } z[0] = 0.5 * M_PI * v[0]; z[1] = 0.5 * M_PI * v[1]; ep = exp (z[1]); em = exp (-z[1]); sincos (z[0], &sx, &cx); s[0] = +0.5 * sx * (em + ep); s[1] = -0.5 * cx * (em - ep); c[0] = +0.5 * cx * (em + ep); c[1] = +0.5 * sx * (em - ep); w = (0.5 + v[0])*(0.5 + v[0]) - v[1] * v[1]; z[1] = v[1]; while (v[0] <= 6.0) { v[0] += 2.0; z[0] = v[0] - 1.0; Cdiv (z, v, tmp); Cmul (R,tmp,r); R[0] = r[0]; R[1] = r[1]; } z[0] = v[0] + 1.0; tmp[0] = 0.25; tmp[1] = 0.0; Cdiv (tmp, z, X); tmp[0] = 0.35 + 6.1 * X[0]; tmp[1] = 6.1*X[1]; Cmul (X, tmp, z); z[0] = 1.0 - 3.0*z[0]; z[1] = -3.0*z[1]; Cmul (X, z, tmp); G[0] = 1.0 + 5.0 * tmp[0]; G[1] = 5.0 * tmp[1]; z[0] = 8.0 * X[0]; z[1] = 8.0 * X[1]; M = sqrt(hypot(z[0], z[1])); L = 0.5 * atan2 (z[1], z[0]); sincos (L, &sL, &cL); tmp[0] = M * cL; tmp[1] = M * sL; Cmul (G, X, z); z[0] = 1.0 - 0.5*z[0]; z[1] = -0.5*z[1]; Cmul (X, z, r); r[0] = 1.0 - r[0]; r[1] = -r[1]; Cmul (R, r, z); Cdiv (z, tmp, R); u = g = 2.0 * x; f = t = 1.0; k = 0.5; x2 = x * x; Xn = 1.0 + (1e8/(1.0 - x2)); k1 = k + 1.0; t = t * x2 * (k*k - w) / (k1*k1 - 0.25); k += 1.0; f += t; k1 = k + 1.0; u = u * x2 * (k*k - w) / (k1*k1 - 0.25); k += 1.0; g += u; while (k < K || fabs (Xn*t) > fabs(f)) { (*iter)++; k1 = k + 1.0; t = t * x2 * (k*k - w) / (k1*k1 - 0.25); k += 1.0; k1 = k + 1.0; f += t; u = u * x2 * (k*k - w) / (k1*k1 - 0.25); k += 1.0; g += u; } L = x2 / (1.0 - x2); f += (t * L); g += (u * L); Cmul(s,R,z); Cdiv(c,R,tmp); pq[0] = (g*z[0] + f*tmp[0])/M_SQRT_PI; pq[1] = (g*z[1] + f*tmp[1])/M_SQRT_PI; Cmul(c,R,z); Cdiv(s,R,tmp); pq[2] = a[0]*M_SQRT_PI*(g*z[0] - f*tmp[0])/2.0; pq[3] = a[1]*M_SQRT_PI*(g*z[1] - f*tmp[1])/2.0; }