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libdspl-2.0/dspl/src/math.c

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/*
* Copyright (c) 2015-2020 Sergey Bakhurin
* Digital Signal Processing Library [http://dsplib.org]
*
* This file is part of DSPL.
*
* 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, either version 3 of the License, or
* (at your option) any later version.
*
* DSPL 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.
*
* You should have received a copy of the GNU General Public License
* along with Foobar. If not, see <http://www.gnu.org/licenses/>.
*/
#include <stdio.h>
#include <stdlib.h>
#include "dspl.h"
/******************************************************************************
\ingroup SPEC_MATH_TRIG_GROUP
\fn int acos_cmplx(complex_t* x, int n, complex_t *y)
\brief The inverse of the cosine function the complex vector argument `x`
Function calculates the inverse of the cosine function as: \n
\f[
\textrm{Arccos}(x) = \frac{\pi}{2} - \textrm{Arcsin}(x) =
\frac{\pi}{2} -j \textrm{Ln}\left( j x + \sqrt{1 - x^2} \right)
\f]
\param[in] x Pointer to the argument vector `x`. \n
Vector size is `[n x 1]`. \n \n
\param[in] n Input vector `x` and the inverse cosine vector `y` size. \n \n
\param[out] y Pointer to the output complex vector `y`,
corresponds to the input vector `x`. \n
Vector size is `[n x 1]`. \n
Memory must be allocated. \n \n
\return
`RES_OK` if function calculated successfully. \n
Else \ref ERROR_CODE_GROUP "code error". \n
Example: \n
\code{.cpp}
complex_t x[3] = {{1.0, 2.0}, {3.0, 4.0}, {5.0, 6.0}};
complex_t y[3];
int k;
acos_cmplx(x, 3, y);
for(k = 0; k < 3; k++)
printf("acos_cmplx(%.1f%+.1fj) = %.3f%+.3fj\n",
RE(x[k]), IM(x[k]), RE(y[k]), IM(y[k]));
\endcode
\n
Output is: \n
\verbatim
acos_cmplx(1.0+2.0j) = 1.144-1.529j
acos_cmplx(3.0+4.0j) = 0.937-2.306j
acos_cmplx(5.0+6.0j) = 0.880-2.749j
\endverbatim
\author
Sergey Bakhurin www.dsplib.org
*******************************************************************************/
int DSPL_API acos_cmplx(complex_t* x, int n, complex_t *y)
{
int k, res;
double pi2 = 0.5 * M_PI;
res = asin_cmplx(x, n, y);
if(res != RES_OK)
return res;
for(k = 0; k < n; k++)
{
RE(y[k]) = pi2 - RE(y[k]);
IM(y[k]) = - IM(y[k]);
}
return RES_OK;
}
/******************************************************************************
\ingroup SPEC_MATH_TRIG_GROUP
\fn int asin_cmplx(complex_t* x, int n, complex_t *y)
\brief The inverse of the sine function the complex vector argument `x`
Function calculates the inverse of the sine function as: \n
\f[
\textrm{Arcsin}(x) = j \textrm{Ln}\left( j x + \sqrt{1 - x^2} \right)
\f]
\param[in] x Pointer to the argument vector `x`. \n
Vector size is `[n x 1]`. \n \n
\param[in] n Input vector `x` and the inverse sine vector `y` size. \n \n
\param[out] y Pointer to the output complex vector `y`,
corresponds to the input vector `x`. \n
Vector size is `[n x 1]`. \n
Memory must be allocated. \n \n
\return
`RES_OK` if function calculated successfully. \n
Else \ref ERROR_CODE_GROUP "code error". \n
Example: \n
\code{.cpp}
complex_t x[3] = {{1.0, 2.0}, {3.0, 4.0}, {5.0, 6.0}};
complex_t y[3];
int k;
asin_cmplx(x, 3, y);
for(k = 0; k < 3; k++)
printf("asin_cmplx(%.1f%+.1fj) = %.3f%+.3fj\n",
RE(x[k]), IM(x[k]), RE(y[k]), IM(y[k]));
\endcode
\n
Output is: \n
\verbatim
asin_cmplx(1.0+2.0j) = 0.427+1.529j
asin_cmplx(3.0+4.0j) = 0.634+2.306j
asin_cmplx(5.0+6.0j) = 0.691+2.749j
\endverbatim
\author
Sergey Bakhurin www.dsplib.org
*******************************************************************************/
int DSPL_API asin_cmplx(complex_t* x, int n, complex_t *y)
{
int k;
complex_t tmp;
if(!x || !y)
return ERROR_PTR;
if(n < 1)
return ERROR_SIZE;
for(k = 0; k < n; k++)
{
RE(tmp) = 1.0 - CMRE(x[k], x[k]); /* 1-x[k]^2 */
IM(tmp) = - CMIM(x[k], x[k]); /* 1-x[k]^2 */
sqrt_cmplx(&tmp, 1, y+k); /* sqrt(1 - x[k]^2) */
RE(y[k]) -= IM(x[k]); /* j * x[k] + sqrt(1 - x[k]^2) */
IM(y[k]) += RE(x[k]); /* j * x[k] + sqrt(1 - x[k]^2) */
log_cmplx(y+k, 1, &tmp); /* log( j * x[k] + sqrt(1 - x[k]^2) ) */
RE(y[k]) = IM(tmp); /* -j * log( j * x[k] + sqrt(1 - x[k]^2) ) */
IM(y[k]) = -RE(tmp); /* -j * log( j * x[k] + sqrt(1 - x[k]^2) ) */
}
return RES_OK;
}
/*******************************************************************************
Modified Bessel Function of the First Kind – I0(x) [1]
[1] Rational Approximations for the Modified Bessel Function
of the First Kind – I0(x) for Computations with Double Precision
by PAVEL HOLOBORODKO on NOVEMBER 11, 2015
https://www.advanpix.com/2015/11/11/
*******************************************************************************/
int DSPL_API bessel_i0(double* x, int n, double* y)
{
double P16[17] = { 1.0000000000000000000000801e+00,
2.4999999999999999999629693e-01,
2.7777777777777777805664954e-02,
1.7361111111111110294015271e-03,
6.9444444444444568581891535e-05,
1.9290123456788994104574754e-06,
3.9367598891475388547279760e-08,
6.1511873265092916275099070e-10,
7.5940584360755226536109511e-12,
7.5940582595094190098755663e-14,
6.2760839879536225394314453e-16,
4.3583591008893599099577755e-18,
2.5791926805873898803749321e-20,
1.3141332422663039834197910e-22,
5.9203280572170548134753422e-25,
2.0732014503197852176921968e-27,
1.1497640034400735733456400e-29};
double P22[23] = { 3.9894228040143265335649948e-01,
4.9867785050353992900698488e-02,
2.8050628884163787533196746e-02,
2.9219501690198775910219311e-02,
4.4718622769244715693031735e-02,
9.4085204199017869159183831e-02,
-1.0699095472110916094973951e-01,
2.2725199603010833194037016e+01,
-1.0026890180180668595066918e+03,
3.1275740782277570164423916e+04,
-5.9355022509673600842060002e+05,
2.6092888649549172879282592e+06,
2.3518420447411254516178388e+08,
-8.9270060370015930749184222e+09,
1.8592340458074104721496236e+11,
-2.6632742974569782078420204e+12,
2.7752144774934763122129261e+13,
-2.1323049786724612220362154e+14,
1.1989242681178569338129044e+15,
-4.8049082153027457378879746e+15,
1.3012646806421079076251950e+16,
-2.1363029690365351606041265e+16,
1.6069467093441596329340754e+16};
double x2;
int k;
if(!x || !y)
return ERROR_PTR;
if(n < 1)
return ERROR_SIZE;
for(k =0; k < n; k++)
{
if(x[k] < 0.0)
return ERROR_NEGATIVE;
if(x[k] < 7.75)
{
x2 = x[k] * x[k] * 0.25;
polyval(P16, 16, &x2, 1, y+k);
y[k] = x2 * y[k] + 1.0;
}
else
{
x2 = 1.0 / x[k];
polyval(P22, 22, &x2, 1, y+k);
y[k] *= exp(x[k]) / sqrt(x[k]);
}
}
return RES_OK;
}
/*******************************************************************************
module operator for double
*******************************************************************************/
double DSPL_API dmod (double x, double y)
{
if(y == 0.0)
return x;
return x - floor(x/y) * y;
}
/******************************************************************************
\ingroup SPEC_MATH_TRIG_GROUP
\fn int cos_cmplx(complex_t* x, int n, complex_t *y)
\brief The cosine function the complex vector argument `x`
Function calculates the cosine function as: \n
\f[
\textrm{cos}(x) = \frac{\exp(jx) + \exp(-jx)}{2}
\f]
\param[in] x Pointer to the argument vector `x`. \n
Vector size is `[n x 1]`. \n \n
\param[in] n Input vector `x` and the cosine vector `y` size. \n \n
\param[out] y Pointer to the output complex vector `y`,
corresponds to the input vector `x`. \n
Vector size is `[n x 1]`. \n
Memory must be allocated. \n \n
\return
`RES_OK` if function calculated successfully. \n
Else \ref ERROR_CODE_GROUP "code error". \n
Example: \n
\code{.cpp}
complex_t x[3] = {{1.0, 2.0}, {3.0, 4.0}, {5.0, 6.0}};
complex_t y[3];
int k;
cos_cmplx(x, 3, y);
for(k = 0; k < 3; k++)
printf("cos_cmplx(%.1f%+.1fj) = %9.3f%+9.3fj\n",
RE(x[k]), IM(x[k]), RE(y[k]), IM(y[k]));
\endcode
\n
Output is: \n
\verbatim
cos_cmplx(1.0+2.0j) = 2.033 -3.052j
cos_cmplx(3.0+4.0j) = -27.035 -3.851j
cos_cmplx(5.0+6.0j) = 57.219 +193.428j
\endverbatim
\author
Sergey Bakhurin www.dsplib.org
*******************************************************************************/
int DSPL_API cos_cmplx(complex_t* x, int n, complex_t *y)
{
int k;
double ep, em, sx, cx;
if(!x || !y)
return ERROR_PTR;
if(n < 1)
return ERROR_SIZE;
for(k = 0; k < n; k++)
{
ep = exp( IM(x[k]));
em = exp(-IM(x[k]));
sx = 0.5 * sin(RE(x[k]));
cx = 0.5 * cos(RE(x[k]));
RE(y[k]) = cx * (em + ep);
IM(y[k]) = sx * (em - ep);
}
return RES_OK;
}
/******************************************************************************
\ingroup SPEC_MATH_COMMON_GROUP
\fn int log_cmplx(complex_t* x, int n, complex_t *y)
\brief The logarithm function the complex vector argument `x`
Function calculates the logarithm function as: \n
\f[
\textrm{Ln}(x) = j \varphi + \ln(|x|),
\f]
here \f$\varphi\f$ - the complex number phase.
\param[in] x Pointer to the argument vector `x`. \n
Vector size is `[n x 1]`. \n \n
\param[in] n Input vector `x` and the logarithm vector `y` size. \n \n
\param[out] y Pointer to the output complex vector `y`,
corresponds to the input vector `x`. \n
Vector size is `[n x 1]`. \n
Memory must be allocated. \n \n
\return
`RES_OK` if function calculated successfully. \n
Else \ref ERROR_CODE_GROUP "code error". \n
Example: \n
\code{.cpp}
complex_t x[3] = {{1.0, 2.0}, {3.0, 4.0}, {5.0, 6.0}};
complex_t y[3];
int k;
log_cmplx(x, 3, y);
for(k = 0; k < 3; k++)
printf("log_cmplx(%.1f%+.1fj) = %.3f%+.3fj\n",
RE(x[k]), IM(x[k]), RE(y[k]), IM(y[k]));
\endcode
\n
Output is: \n
\verbatim
log_cmplx(1.0+2.0j) = 0.805+1.107j
log_cmplx(3.0+4.0j) = 1.609+0.927j
log_cmplx(5.0+6.0j) = 2.055+0.876j
\endverbatim
\author
Sergey Bakhurin www.dsplib.org
*******************************************************************************/
int DSPL_API log_cmplx(complex_t* x, int n, complex_t *y)
{
int k;
if(!x || !y)
return ERROR_PTR;
if(n < 1)
return ERROR_SIZE;
for(k = 0; k < n; k++)
{
RE(y[k]) = 0.5 * log(ABSSQR(x[k]));
IM(y[k]) = atan2(IM(x[k]), RE(x[k]));
}
return RES_OK;
}
/******************************************************************************
\ingroup SPEC_MATH_TRIG_GROUP
\fn int sin_cmplx(complex_t* x, int n, complex_t *y)
\brief The sine function the complex vector argument `x`
Function calculates the sine function as: \n
\f[
\textrm{cos}(x) = \frac{\exp(jx) - \exp(-jx)}{2j}
\f]
\param[in] x Pointer to the argument vector `x`. \n
Vector size is `[n x 1]`. \n \n
\param[in] n Input vector `x` and the sine vector `y` size. \n \n
\param[out] y Pointer to the output complex vector `y`,
corresponds to the input vector `x`. \n
Vector size is `[n x 1]`. \n
Memory must be allocated. \n \n
\return
`RES_OK` if function calculated successfully. \n
Else \ref ERROR_CODE_GROUP "code error". \n
Example: \n
\code{.cpp}
complex_t x[3] = {{1.0, 2.0}, {3.0, 4.0}, {5.0, 6.0}};
complex_t y[3];
int k;
sin_cmplx(x, 3, y);
for(k = 0; k < 3; k++)
printf("sin_cmplx(%.1f%+.1fj) = %9.3f%+9.3fj\n",
RE(x[k]), IM(x[k]), RE(y[k]), IM(y[k]));
\endcode
\n
Output is: \n
\verbatim
sin_cmplx(1.0+2.0j) = 3.166 +1.960j
sin_cmplx(3.0+4.0j) = 3.854 -27.017j
sin_cmplx(5.0+6.0j) = -193.430 +57.218j
\endverbatim
\author
Sergey Bakhurin www.dsplib.org
*******************************************************************************/
int DSPL_API sin_cmplx(complex_t* x, int n, complex_t *y)
{
int k;
double ep, em, sx, cx;
if(!x || !y)
return ERROR_PTR;
if(n < 1)
return ERROR_SIZE;
for(k = 0; k < n; k++)
{
ep = exp( IM(x[k]));
em = exp(-IM(x[k]));
sx = 0.5 * sin(RE(x[k]));
cx = 0.5 * cos(RE(x[k]));
RE(y[k]) = sx * (em + ep);
IM(y[k]) = cx * (ep - em);
}
return RES_OK;
}
/*******************************************************************************
sinc(x) = sin(pi*x)/(pi*x)
*******************************************************************************/
int DSPL_API sinc(double* x, int n, double a, double* y)
{
int k;
if(!x || !y)
return ERROR_PTR;
if(n<1)
return ERROR_SIZE;
for(k = 0; k < n; k++)
y[k] = (x[k]==0.0) ? 1.0 : sin(a*x[k])/(a*x[k]);
return RES_OK;
}
/*******************************************************************************
Sine integral
--------------------------------------------------------------------------------
This function uses Padé approximants of the convergent Taylor series [1]
[1]
https://www.sciencedirect.com/science/article/pii/S221313371500013X?via%3Dihub
*******************************************************************************/
int DSPL_API sine_int(double* x, int n, double* si)
{
int k, sgn, p;
double num, den, y, x2, x22, z, f, g;
double A[8] = {+1.00000000000000000E0 ,
-4.54393409816329991E-2 ,
+1.15457225751016682E-3 ,
-1.41018536821330254E-5 ,
+9.43280809438713025E-8 ,
-3.53201978997168357E-10 ,
+7.08240282274875911E-13 ,
-6.05338212010422477E-16 };
double B[7] = {+1.0 ,
+1.01162145739225565E-2 ,
+4.99175116169755106E-5 ,
+1.55654986308745614E-7 ,
+3.28067571055789734E-10 ,
+4.50490975753865810E-13 ,
+3.21107051193712168E-16 };
double FA[11] = {+1.000000000000000000000E0,
+7.444370681619367006180E2,
+1.963963728951468698010E5,
+2.377503101254318340340E7,
+1.430734038212746368880E9,
+4.33736238870432522765E10,
+6.40533830574022022911E11,
+4.20968180571076940208E12,
+1.00795182980368574617E13,
+4.94816688199951963482E12,
-4.94701168645415959931E11};
double FB[10] = {+1.000000000000000000000E0,
+7.464370681619276780310E2,
+1.978652470315839514500E5,
+2.415356701651268451440E7,
+1.474789521929854649580E9,
+4.58595115847765779830E10,
+7.08501308149515401563E11,
+5.06084464593475076774E12,
+1.43468549171581016479E13,
+1.11535493509914254097E13};
double GA[11] = {+1.000000000000000000E0,
+8.135952011516861500E2,
+2.352391816264782000E5,
+3.125575707957787310E7,
+2.062975951467633540E9,
+6.83052205423625007E10,
+1.09049528450362786E12,
+7.57664583257834349E12,
+1.81004487464664575E13,
+6.43291613143049485E12,
-1.36517137670871689E12};
double GB[10] = {+1.000000000000000000E0,
+8.195952011514515640E2,
+2.400367528355787770E5,
+3.260266616470908220E7,
+2.233555432780993600E9,
+7.87465017341829930E10,
+1.39866710696414565E12,
+1.17164723371736605E13,
+4.01839087307656620E13,
+3.99653257887490811E13};
if(!x || !si)
return ERROR_PTR;
if(n<1)
return ERROR_SIZE;
for(p = 0; p < n; p++)
{
sgn = x[p] > 0.0 ? 0 : 1;
y = x[p] < 0.0 ? -x[p] : x[p];
if(y < 4)
{
x2 = y * y;
z = 1.0;
num = 0.0;
for(k = 0; k < 8; k++)
{
num += A[k] * z;
z*=x2;
}
z = 1.0;
den = 0.0;
for(k = 0; k < 7; k++)
{
den += B[k]*z;
z*=x2;
}
si[p] = x[p] * num/den;
}
else
{
x2 = 1.0/y;
x22 = x2*x2;
z = 1.0;
num = 0.0;
for(k = 0; k < 11; k++)
{
num += FA[k] * z;
z*=x22;
}
z = 1.0;
den = 0.0;
for(k = 0; k < 10; k++)
{
den += FB[k]*z;
z*=x22;
}
f = x2 * num / den;
z = 1.0;
num = 0.0;
for(k = 0; k < 11; k++)
{
num += GA[k] * z;
z*=x22;
}
z = 1.0;
den = 0.0;
for(k = 0; k < 10; k++)
{
den += GB[k]*z;
z*=x22;
}
g = x22 * num / den;
si[p] = sgn ? f * cos(y) + g * sin(y) - M_PI * 0.5 :
M_PI * 0.5 - f * cos(y) - g * sin(y);
}
}
return RES_OK;
}
/******************************************************************************
\ingroup SPEC_MATH_COMMON_GROUP
\fn int sqrt_cmplx(complex_t* x, int n, complex_t *y)
\brief Square root of the complex vector argguument `x`.
Function calculates square root value of vector `x` length `n`: \n
\f[
y(k) = \sqrt{x(k)}, \qquad k = 0 \ldots n-1.
\f]
\param[in] x Pointer to the input complex vector `x`. \n
Vector size is `[n x 1]`. \n \n
\param[in] n Size of input and output vectors `x` and `y`. \n \n
\param[out] y Pointer to the square root vector `y`. \n
Vector size is `[n x 1]`. \n
Memory must be allocated. \n \n
\return `RES_OK` if function is calculated successfully. \n
Else \ref ERROR_CODE_GROUP "code error". \n
Example
\code{.cpp}
complex_t x[3] = {{1.0, 2.0}, {3.0, 4.0}, {5.0, 6.0}};
complex_t y[3]
int k;
sqrt_cmplx(x, 3, y);
for(k = 0; k < 3; k++)
printf("sqrt_cmplx(%.1f%+.1fj) = %.3f%+.3fj\n",
RE(x[k]), IM(x[k]), RE(y[k]), IM(y[k]));
\endcode
\n
Результатом работы будет
\verbatim
sqrt_cmplx(1.0+2.0j) = 1.272+0.786j
sqrt_cmplx(3.0+4.0j) = 2.000+1.000j
sqrt_cmplx(5.0+6.0j) = 2.531+1.185j
\endverbatim
\author Sergey Bakhurin www.dsplib.org
*******************************************************************************/
int DSPL_API sqrt_cmplx(complex_t* x, int n, complex_t *y)
{
int k;
double r, zr, at;
complex_t t;
if(!x || !y)
return ERROR_PTR;
if(n < 1)
return ERROR_SIZE;
for(k = 0; k < n; k++)
{
r = ABS(x[k]);
if(r == 0.0)
{
RE(y[k]) = 0.0;
IM(y[k]) = 0.0;
}
else
{
RE(t) = RE(x[k]) + r;
IM(t) = IM(x[k]);
at = ABS(t);
if(at == 0.0)
{
RE(y[k]) = 0.0;
IM(y[k]) = sqrt(r);
}
else
{
zr = 1.0 / ABS(t);
r = sqrt(r);
RE(y[k]) = RE(t) * zr * r;
IM(y[k]) = IM(t) * zr * r;
}
}
}
return RES_OK;
}
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