/* GSL - Generic Sound Layer * Copyright (C) 2001 Stefan Westerfeld and Tim Janik * * This library is free software; you can redistribute it and/or * modify it under the terms of the GNU Lesser General Public * License as published by the Free Software Foundation; either * version 2 of the License, or (at your option) any later version. * * This library 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 * Lesser General Public License for more details. * * You should have received a copy of the GNU Lesser General * Public License along with this library; if not, write to the * Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, * Boston, MA 02110-1301, USA. */ #ifndef __GSL_MATH_H__ #define __GSL_MATH_H__ #include #include #include #ifdef __cplusplus extern "C" { #endif /* __cplusplus */ /* --- constants --- */ #define GSL_PI (3.1415926535897932384626433832795029 /* pi */) #define GSL_PI_DIV_2 (1.5707963267948966192313216916397514 /* pi/2 */) #define GSL_PI_DIV_4 (0.7853981633974483096156608458198757 /* pi/4 */) #define GSL_1_DIV_PI (0.3183098861837906715377675267450287 /* 1/pi */) #define GSL_2_DIV_PI (0.6366197723675813430755350534900574 /* 2/pi */) #define GSL_2_DIV_SQRT_PI (1.1283791670955125738961589031215452 /* 2/sqrt(pi) */) #define GSL_SQRT2 (1.4142135623730950488016887242096981 /* sqrt(2) */) #define GSL_1_DIV_SQRT2 (0.7071067811865475244008443621048490 /* 1/sqrt(2) */) #define GSL_E (2.7182818284590452353602874713526625 /* e */) #define GSL_LOG2E (1.4426950408889634073599246810018922 /* log_2(e) */) #define GSL_LOG10E (0.4342944819032518276511289189166051 /* log_10(e) */) #define GSL_LN2 (0.6931471805599453094172321214581766 /* ln(2) */) #define GSL_LN10 (2.3025850929940456840179914546843642 /* ln(10) */) #define GSL_2_POW_1_DIV_12 (1.0594630943592952645618252949463417 /* 2^(1/12) */) #define GSL_2_POW_1_DIV_72 (1.0096735332285108621925214011186051 /* 2^(1/72) */) #define GSL_LN_2_POW_1_DIV_12 (0.0577622650466621091181026767881814 /* ln(2^(1/12)) */) #define GSL_LN_2_POW_1_DIV_72 (0.0096270441744436848530171127980302 /* ln(2^(1/72)) */) #define GSL_LOG2_10 (3.3219280948873623478703194294893902 /* log_2(10) */) #define GSL_LOG2POW20_10 (0.1660964047443681173935159714744695 /* log_2(10)/20 */) /* --- structures --- */ struct _GslComplex { double re; double im; }; /* --- float/double signbit extraction --- */ #ifdef signbit # define gsl_float_sign(dblflt) (signbit (dblflt)) #else # define gsl_float_sign(dblflt) ((dblflt) < -0.0) /* good enough for us */ #endif /* --- complex numbers --- */ static inline GslComplex gsl_complex (double re, double im); static inline GslComplex gsl_complex_polar (double abs, double arg); static inline GslComplex gsl_complex_add (GslComplex c1, GslComplex c2); static inline GslComplex gsl_complex_add3 (GslComplex c1, GslComplex c2, GslComplex c3); static inline GslComplex gsl_complex_sub (GslComplex c1, GslComplex c2); static inline GslComplex gsl_complex_sub3 (GslComplex c1, GslComplex c2, GslComplex c3); static inline GslComplex gsl_complex_scale (GslComplex c1, double scale); static inline GslComplex gsl_complex_mul (GslComplex c1, GslComplex c2); static inline GslComplex gsl_complex_mul3 (GslComplex c1, GslComplex c2, GslComplex c3); static inline GslComplex gsl_complex_div (GslComplex a, GslComplex b); static inline GslComplex gsl_complex_reciprocal (GslComplex c); static inline GslComplex gsl_complex_sqrt (GslComplex z); static inline GslComplex gsl_complex_conj (GslComplex c); /* {re, -im} */ static inline GslComplex gsl_complex_id (GslComplex c); static inline GslComplex gsl_complex_inv (GslComplex c); /* {-re, -im} */ static inline double gsl_complex_abs (GslComplex c); static inline double gsl_complex_arg (GslComplex c); static inline GslComplex gsl_complex_sin (GslComplex c); static inline GslComplex gsl_complex_cos (GslComplex c); static inline GslComplex gsl_complex_tan (GslComplex c); static inline GslComplex gsl_complex_sinh (GslComplex c); static inline GslComplex gsl_complex_cosh (GslComplex c); static inline GslComplex gsl_complex_tanh (GslComplex c); char* gsl_complex_str (GslComplex c); char* gsl_complex_list (unsigned int n_points, GslComplex *points, const char *indent); void gsl_complex_gnuplot (const char *file_name, unsigned int n_points, GslComplex *points); /* --- polynomials --- */ /* example, degree=2: 5+2x+7x^2 => a[0..degree] = { 5, 2, 7 } */ static inline void gsl_poly_add (unsigned int degree, double *a, /* a[0..degree] */ double *b); static inline void gsl_poly_sub (unsigned int order, double *a, /* [0..degree] */ double *b); static inline void gsl_poly_mul (double *p, /* out:[0..aorder+border] */ unsigned int aorder, const double *a, /* in:[0..aorder] */ unsigned int border, const double *b); /* in:[0..border] */ static inline void gsl_poly_scale (unsigned int order, double *a, /* [0..degree] */ double scale); static inline void gsl_poly_xscale (unsigned int order, double *a, /* [0..degree] */ double xscale); static inline double gsl_poly_eval (unsigned int degree, double *a, /* [0..degree] */ double x); void gsl_poly_complex_roots (unsigned int poly_degree, double *a, /* [0..degree] (degree+1 elements) */ GslComplex *roots); /* [degree] */ void gsl_poly_from_re_roots (unsigned int poly_degree, double *a, /* [0..degree] */ GslComplex *roots); void gsl_cpoly_from_roots (unsigned int poly_degree, GslComplex *c, /* [0..degree] */ GslComplex *roots); static inline void gsl_cpoly_mul_monomial (unsigned int degree, /* _new_ degree */ GslComplex *c, /* in:[0..degree-1] out:[0..degree] */ GslComplex root); /* c(x) *= (x^1 - root) */ static inline void gsl_cpoly_mul_reciprocal (unsigned int degree, /* _new_ degree */ GslComplex *c, /* in:[0..degree-1] out:[0..degree] */ GslComplex root); /* c(x) *= (1 - root * x^-1) */ static inline void gsl_cpoly_mul (GslComplex *p, /* out:[0..aorder+border] */ unsigned int aorder, GslComplex *a, /* in:[0..aorder] */ unsigned int border, GslComplex *b); /* in:[0..border] */ char* gsl_poly_str (unsigned int degree, double *a, const char *var); char* gsl_poly_str1 (unsigned int degree, double *a, const char *var); /* --- transformations --- */ double gsl_temp_freq (double kammer_freq, int halftone_delta); /* --- miscellaneous --- */ double gsl_bit_depth_epsilon (guint n_bits); /* 1..32 */ /* --- ellipses --- */ double gsl_ellip_rf (double x, double y, double z); double gsl_ellip_F (double phi, double ak); double gsl_ellip_sn (double u, double emmc); double gsl_ellip_asn (double y, double emmc); GslComplex gsl_complex_ellip_asn (GslComplex y, GslComplex emmc); GslComplex gsl_complex_ellip_sn (GslComplex u, GslComplex emmc); /* --- implementations --- */ static inline GslComplex gsl_complex (double re, double im) { GslComplex r; r.re = re; r.im = im; return r; } static inline GslComplex gsl_complex_polar (double abs, double arg) { return gsl_complex (abs * cos (arg), abs * sin (arg)); } static inline GslComplex gsl_complex_add (GslComplex c1, GslComplex c2) { return gsl_complex (c1.re + c2.re, c1.im + c2.im); } static inline GslComplex gsl_complex_add3 (GslComplex c1, GslComplex c2, GslComplex c3) { return gsl_complex (c1.re + c2.re + c3.re, c1.im + c2.im + c3.im); } static inline GslComplex gsl_complex_sub (GslComplex c1, GslComplex c2) { return gsl_complex (c1.re - c2.re, c1.im - c2.im); } static inline GslComplex gsl_complex_sub3 (GslComplex c1, GslComplex c2, GslComplex c3) { return gsl_complex (c1.re - c2.re - c3.re, c1.im - c2.im - c3.im); } static inline GslComplex gsl_complex_scale (GslComplex c1, double scale) { return gsl_complex (c1.re * scale, c1.im * scale); } static inline GslComplex gsl_complex_mul (GslComplex c1, GslComplex c2) { return gsl_complex (c1.re * c2.re - c1.im * c2.im, c1.re * c2.im + c1.im * c2.re); } static inline GslComplex gsl_complex_mul3 (GslComplex c1, GslComplex c2, GslComplex c3) { double aec = c1.re * c2.re * c3.re; double bde = c1.im * c2.im * c3.re; double adf = c1.re * c2.im * c3.im; double bcf = c1.im * c2.re * c3.im; double ade = c1.re * c2.im * c3.re; double bce = c1.im * c2.re * c3.re; double acf = c1.re * c2.re * c3.im; double bdf = c1.im * c2.im * c3.im; return gsl_complex (aec - bde - adf - bcf, ade + bce + acf - bdf); } static inline GslComplex gsl_complex_div (GslComplex a, GslComplex b) { GslComplex c; if (fabs (b.re) >= fabs (b.im)) { double r = b.im / b.re, den = b.re + r * b.im; c.re = (a.re + r * a.im) / den; c.im = (a.im - r * a.re) / den; } else { double r = b.re / b.im, den = b.im + r * b.re; c.re = (a.re * r + a.im) / den; c.im = (a.im * r - a.re) / den; } return c; } static inline GslComplex gsl_complex_reciprocal (GslComplex c) { if (fabs (c.re) >= fabs (c.im)) { double r = c.im / c.re, den = c.re + r * c.im; c.re = 1. / den; c.im = - r / den; } else { double r = c.re / c.im, den = c.im + r * c.re; c.re = r / den; c.im = - 1. / den; } return c; } static inline GslComplex gsl_complex_sqrt (GslComplex z) { if (z.re == 0.0 && z.im == 0.0) return z; else { GslComplex c; double w, x = fabs (z.re), y = fabs (z.im); if (x >= y) { double r = y / x; w = sqrt (x) * sqrt (0.5 * (1.0 + sqrt (1.0 + r * r))); } else { double r = x / y; w = sqrt (y) * sqrt (0.5 * (r + sqrt (1.0 + r * r))); } if (z.re >= 0.0) { c.re = w; c.im = z.im / (2.0 * w); } else { c.im = z.im >= 0 ? w : -w; c.re = z.im / (2.0 * c.im); } return c; } } static inline GslComplex gsl_complex_conj (GslComplex c) { return gsl_complex (c.re, -c.im); } static inline GslComplex gsl_complex_inv (GslComplex c) { return gsl_complex (-c.re, -c.im); } static inline GslComplex gsl_complex_id (GslComplex c) { return c; } static inline double gsl_complex_abs (GslComplex c) { /* compute (a^2 + b^2)^(1/2) without destructive underflow or overflow */ double absa = fabs (c.re), absb = fabs (c.im); return (absa > absb ? absb == 0.0 ? absa : absa * sqrt (1.0 + (absb / absa) * (absb / absa)) : absb == 0.0 ? 0.0 : absb * sqrt (1.0 + (absa / absb) * (absa / absb))); } static inline double gsl_complex_arg (GslComplex c) { double a = atan2 (c.im, c.re); return a; } static inline GslComplex gsl_complex_sin (GslComplex c) { return gsl_complex (sin (c.re) * cosh (c.im), cos (c.re) * sinh (c.im)); } static inline GslComplex gsl_complex_cos (GslComplex c) { return gsl_complex (cos (c.re) * cosh (c.im), - sin (c.re) * sinh (c.im)); } static inline GslComplex gsl_complex_tan (GslComplex c) { return gsl_complex_div (gsl_complex (tan (c.re), tanh (c.im)), gsl_complex (1.0, -tan (c.re) * tanh (c.im))); } static inline GslComplex gsl_complex_sinh (GslComplex c) { return gsl_complex (sinh (c.re) * cos (c.im), cosh (c.re) * sin (c.im)); } static inline GslComplex gsl_complex_cosh (GslComplex c) { return gsl_complex (cosh (c.re) * cos (c.im), sinh (c.re) * sin (c.im)); } static inline GslComplex gsl_complex_tanh (GslComplex c) { return gsl_complex_div (gsl_complex_sinh (c), gsl_complex_cosh (c)); } static inline void gsl_poly_add (unsigned int degree, double *a, double *b) { unsigned int i; for (i = 0; i <= degree; i++) a[i] += b[i]; } static inline void gsl_poly_sub (unsigned int degree, double *a, double *b) { unsigned int i; for (i = 0; i <= degree; i++) a[i] -= b[i]; } static inline void gsl_poly_mul (double *p, /* out:[0..aorder+border] */ unsigned int aorder, const double *a, /* in:[0..aorder] */ unsigned int border, const double *b) /* in:[0..border] */ { unsigned int i; for (i = aorder + border; i > 0; i--) { unsigned int j; double t = 0; for (j = i - MIN (border, i); j <= MIN (aorder, i); j++) t += a[j] * b[i - j]; p[i] = t; } p[0] = a[0] * b[0]; } static inline void gsl_cpoly_mul_monomial (unsigned int degree, GslComplex *c, GslComplex root) { unsigned int j; c[degree] = c[degree - 1]; for (j = degree - 1; j >= 1; j--) c[j] = gsl_complex_sub (c[j - 1], gsl_complex_mul (c[j], root)); c[0] = gsl_complex_mul (c[0], gsl_complex_inv (root)); } static inline void gsl_cpoly_mul_reciprocal (unsigned int degree, GslComplex *c, GslComplex root) { unsigned int j; c[degree] = gsl_complex_mul (c[degree - 1], gsl_complex_inv (root)); for (j = degree - 1; j >= 1; j--) c[j] = gsl_complex_sub (c[j], gsl_complex_mul (c[j - 1], root)); /* c[0] = c[0]; */ } static inline void gsl_cpoly_mul (GslComplex *p, /* [0..aorder+border] */ unsigned int aorder, GslComplex *a, unsigned int border, GslComplex *b) { unsigned int i; for (i = aorder + border; i > 0; i--) { GslComplex t; unsigned int j; t = gsl_complex (0, 0); for (j = i - MIN (i, border); j <= MIN (aorder, i); j++) t = gsl_complex_add (t, gsl_complex_mul (a[j], b[i - j])); p[i] = t; } p[0] = gsl_complex_mul (a[0], b[0]); } static inline void gsl_poly_scale (unsigned int degree, double *a, double scale) { unsigned int i; for (i = 0; i <= degree; i++) a[i] *= scale; } static inline void gsl_poly_xscale (unsigned int degree, double *a, double xscale) { double scale = xscale; unsigned int i; for (i = 1; i <= degree; i++) { a[i] *= scale; scale *= xscale; } } static inline double gsl_poly_eval (unsigned int degree, double *a, double x) { double sum = a[degree]; while (degree--) sum = sum * x + a[degree]; return sum; } #ifdef __cplusplus } #endif /* __cplusplus */ #endif /* __GSL_MATH_H__ */