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+/*
+ * Compute the natural log of Gamma(x), accurate to 10 decimal places.
+ *
+ * This implementation is based on:
+ *
+ * Pike, M.C., I.D. Hill (1966) Algorithm 291: Logarithm of Gamma function
+ * [S14]. Communications of the ACM 9(9):684.
+ */
+static inline double
+ln_gamma(double x) {
+ double f, z;
+
+ assert(x > 0.0);
+
+ if (x < 7.0) {
+ f = 1.0;
+ z = x;
+ while (z < 7.0) {
+ f *= z;
+ z += 1.0;
+ }
+ x = z;
+ f = -log(f);
+ } else {
+ f = 0.0;
+ }
+
+ z = 1.0 / (x * x);
+
+ return f + (x-0.5) * log(x) - x + 0.918938533204673 +
+ (((-0.000595238095238 * z + 0.000793650793651) * z -
+ 0.002777777777778) * z + 0.083333333333333) / x;
+}
+
+/*
+ * Compute the incomplete Gamma ratio for [0..x], where p is the shape
+ * parameter, and ln_gamma_p is ln_gamma(p).
+ *
+ * This implementation is based on:
+ *
+ * Bhattacharjee, G.P. (1970) Algorithm AS 32: The incomplete Gamma integral.
+ * Applied Statistics 19:285-287.
+ */
+static inline double
+i_gamma(double x, double p, double ln_gamma_p) {
+ double acu, factor, oflo, gin, term, rn, a, b, an, dif;
+ double pn[6];
+ unsigned i;
+
+ assert(p > 0.0);
+ assert(x >= 0.0);
+
+ if (x == 0.0) {
+ return 0.0;
+ }
+
+ acu = 1.0e-10;
+ oflo = 1.0e30;
+ gin = 0.0;
+ factor = exp(p * log(x) - x - ln_gamma_p);
+
+ if (x <= 1.0 || x < p) {
+ /* Calculation by series expansion. */
+ gin = 1.0;
+ term = 1.0;
+ rn = p;
+
+ while (true) {
+ rn += 1.0;
+ term *= x / rn;
+ gin += term;
+ if (term <= acu) {
+ gin *= factor / p;
+ return gin;
+ }
+ }
+ } else {
+ /* Calculation by continued fraction. */
+ a = 1.0 - p;
+ b = a + x + 1.0;
+ term = 0.0;
+ pn[0] = 1.0;
+ pn[1] = x;
+ pn[2] = x + 1.0;
+ pn[3] = x * b;
+ gin = pn[2] / pn[3];
+
+ while (true) {
+ a += 1.0;
+ b += 2.0;
+ term += 1.0;
+ an = a * term;
+ for (i = 0; i < 2; i++) {
+ pn[i+4] = b * pn[i+2] - an * pn[i];
+ }
+ if (pn[5] != 0.0) {
+ rn = pn[4] / pn[5];
+ dif = fabs(gin - rn);
+ if (dif <= acu && dif <= acu * rn) {
+ gin = 1.0 - factor * gin;
+ return gin;
+ }
+ gin = rn;
+ }
+ for (i = 0; i < 4; i++) {
+ pn[i] = pn[i+2];
+ }
+
+ if (fabs(pn[4]) >= oflo) {
+ for (i = 0; i < 4; i++) {
+ pn[i] /= oflo;
+ }
+ }
+ }
+ }
+}
+
+/*
+ * Given a value p in [0..1] of the lower tail area of the normal distribution,
+ * compute the limit on the definite integral from [-inf..z] that satisfies p,
+ * accurate to 16 decimal places.
+ *
+ * This implementation is based on:
+ *
+ * Wichura, M.J. (1988) Algorithm AS 241: The percentage points of the normal
+ * distribution. Applied Statistics 37(3):477-484.
+ */
+static inline double
+pt_norm(double p) {
+ double q, r, ret;
+
+ assert(p > 0.0 && p < 1.0);
+
+ q = p - 0.5;
+ if (fabs(q) <= 0.425) {
+ /* p close to 1/2. */
+ r = 0.180625 - q * q;
+ return q * (((((((2.5090809287301226727e3 * r +
+ 3.3430575583588128105e4) * r + 6.7265770927008700853e4) * r
+ + 4.5921953931549871457e4) * r + 1.3731693765509461125e4) *
+ r + 1.9715909503065514427e3) * r + 1.3314166789178437745e2)
+ * r + 3.3871328727963666080e0) /
+ (((((((5.2264952788528545610e3 * r +
+ 2.8729085735721942674e4) * r + 3.9307895800092710610e4) * r
+ + 2.1213794301586595867e4) * r + 5.3941960214247511077e3) *
+ r + 6.8718700749205790830e2) * r + 4.2313330701600911252e1)
+ * r + 1.0);
+ } else {
+ if (q < 0.0) {
+ r = p;
+ } else {
+ r = 1.0 - p;
+ }
+ assert(r > 0.0);
+
+ r = sqrt(-log(r));
+ if (r <= 5.0) {
+ /* p neither close to 1/2 nor 0 or 1. */
+ r -= 1.6;
+ ret = ((((((((7.74545014278341407640e-4 * r +
+ 2.27238449892691845833e-2) * r +
+ 2.41780725177450611770e-1) * r +
+ 1.27045825245236838258e0) * r +
+ 3.64784832476320460504e0) * r +
+ 5.76949722146069140550e0) * r +
+ 4.63033784615654529590e0) * r +
+ 1.42343711074968357734e0) /
+ (((((((1.05075007164441684324e-9 * r +
+ 5.47593808499534494600e-4) * r +
+ 1.51986665636164571966e-2)
+ * r + 1.48103976427480074590e-1) * r +
+ 6.89767334985100004550e-1) * r +
+ 1.67638483018380384940e0) * r +
+ 2.05319162663775882187e0) * r + 1.0));
+ } else {
+ /* p near 0 or 1. */
+ r -= 5.0;
+ ret = ((((((((2.01033439929228813265e-7 * r +
+ 2.71155556874348757815e-5) * r +
+ 1.24266094738807843860e-3) * r +
+ 2.65321895265761230930e-2) * r +
+ 2.96560571828504891230e-1) * r +
+ 1.78482653991729133580e0) * r +
+ 5.46378491116411436990e0) * r +
+ 6.65790464350110377720e0) /
+ (((((((2.04426310338993978564e-15 * r +
+ 1.42151175831644588870e-7) * r +
+ 1.84631831751005468180e-5) * r +
+ 7.86869131145613259100e-4) * r +
+ 1.48753612908506148525e-2) * r +
+ 1.36929880922735805310e-1) * r +
+ 5.99832206555887937690e-1)
+ * r + 1.0));
+ }
+ if (q < 0.0) {
+ ret = -ret;
+ }
+ return ret;
+ }
+}
+
+/*
+ * Given a value p in [0..1] of the lower tail area of the Chi^2 distribution
+ * with df degrees of freedom, where ln_gamma_df_2 is ln_gamma(df/2.0), compute
+ * the upper limit on the definite integral from [0..z] that satisfies p,
+ * accurate to 12 decimal places.
+ *
+ * This implementation is based on:
+ *
+ * Best, D.J., D.E. Roberts (1975) Algorithm AS 91: The percentage points of
+ * the Chi^2 distribution. Applied Statistics 24(3):385-388.
+ *
+ * Shea, B.L. (1991) Algorithm AS R85: A remark on AS 91: The percentage
+ * points of the Chi^2 distribution. Applied Statistics 40(1):233-235.
+ */
+static inline double
+pt_chi2(double p, double df, double ln_gamma_df_2) {
+ double e, aa, xx, c, ch, a, q, p1, p2, t, x, b, s1, s2, s3, s4, s5, s6;
+ unsigned i;
+
+ assert(p >= 0.0 && p < 1.0);
+ assert(df > 0.0);
+
+ e = 5.0e-7;
+ aa = 0.6931471805;
+
+ xx = 0.5 * df;
+ c = xx - 1.0;
+
+ if (df < -1.24 * log(p)) {
+ /* Starting approximation for small Chi^2. */
+ ch = pow(p * xx * exp(ln_gamma_df_2 + xx * aa), 1.0 / xx);
+ if (ch - e < 0.0) {
+ return ch;
+ }
+ } else {
+ if (df > 0.32) {
+ x = pt_norm(p);
+ /*
+ * Starting approximation using Wilson and Hilferty
+ * estimate.
+ */
+ p1 = 0.222222 / df;
+ ch = df * pow(x * sqrt(p1) + 1.0 - p1, 3.0);
+ /* Starting approximation for p tending to 1. */
+ if (ch > 2.2 * df + 6.0) {
+ ch = -2.0 * (log(1.0 - p) - c * log(0.5 * ch) +
+ ln_gamma_df_2);
+ }
+ } else {
+ ch = 0.4;
+ a = log(1.0 - p);
+ while (true) {
+ q = ch;
+ p1 = 1.0 + ch * (4.67 + ch);
+ p2 = ch * (6.73 + ch * (6.66 + ch));
+ t = -0.5 + (4.67 + 2.0 * ch) / p1 - (6.73 + ch
+ * (13.32 + 3.0 * ch)) / p2;
+ ch -= (1.0 - exp(a + ln_gamma_df_2 + 0.5 * ch +
+ c * aa) * p2 / p1) / t;
+ if (fabs(q / ch - 1.0) - 0.01 <= 0.0) {
+ break;
+ }
+ }
+ }
+ }
+
+ for (i = 0; i < 20; i++) {
+ /* Calculation of seven-term Taylor series. */
+ q = ch;
+ p1 = 0.5 * ch;
+ if (p1 < 0.0) {
+ return -1.0;
+ }
+ p2 = p - i_gamma(p1, xx, ln_gamma_df_2);
+ t = p2 * exp(xx * aa + ln_gamma_df_2 + p1 - c * log(ch));
+ b = t / ch;
+ a = 0.5 * t - b * c;
+ s1 = (210.0 + a * (140.0 + a * (105.0 + a * (84.0 + a * (70.0 +
+ 60.0 * a))))) / 420.0;
+ s2 = (420.0 + a * (735.0 + a * (966.0 + a * (1141.0 + 1278.0 *
+ a)))) / 2520.0;
+ s3 = (210.0 + a * (462.0 + a * (707.0 + 932.0 * a))) / 2520.0;
+ s4 = (252.0 + a * (672.0 + 1182.0 * a) + c * (294.0 + a *
+ (889.0 + 1740.0 * a))) / 5040.0;
+ s5 = (84.0 + 264.0 * a + c * (175.0 + 606.0 * a)) / 2520.0;
+ s6 = (120.0 + c * (346.0 + 127.0 * c)) / 5040.0;
+ ch += t * (1.0 + 0.5 * t * s1 - b * c * (s1 - b * (s2 - b * (s3
+ - b * (s4 - b * (s5 - b * s6))))));
+ if (fabs(q / ch - 1.0) <= e) {
+ break;
+ }
+ }
+
+ return ch;
+}
+
+/*
+ * Given a value p in [0..1] and Gamma distribution shape and scale parameters,
+ * compute the upper limit on the definite integral from [0..z] that satisfies
+ * p.
+ */
+static inline double
+pt_gamma(double p, double shape, double scale, double ln_gamma_shape) {
+ return pt_chi2(p, shape * 2.0, ln_gamma_shape) * 0.5 * scale;
+}
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