Horner Method for polynomial Computation

The Horner algorithm is a fast algorithm named after William George Horner, a British mathematician. In fact, it has been used by Paolo ruffini before. Chinese mathematician Qin Jiuyi proposed this algorithm more than William George Horner 600 years earlier.

P (x) is a polynomial:

We want to calculate the polynomial value p (x0) when X is a special value x0 ).

Construct a sequence:

Then the B0 value of this sequence is the polynomial value.

The program implementation is as follows:

double horner(double p[], int n, double x){    double sum;    sum = p[--n];    while ( n > 0 )    {        sum = p[--n] + sum * x;    }    return sum;}

I often need to design FIR and IIR filters. After designing the filters, I need to verify whether the frequency characteristics are correct. In this case, we need to calculate the result of the real system number Polynomial in the case of X to obtain the complex value. Therefore, the following code is available:

 double _Complex horner_C(double p[], int n, double _Complex x) {    double _Complex sum;    sum = p[--n];    while ( n > 0 )    {        sum = p[--n] + sum * x;    }    return sum; }

This Code uses the new features supported by the plural in c99 and requires the compiler to support the c99 standard.

Finally, a test code is provided. The polynomial P constitutes a FIR low-pass filter, which is designed by using the ffilt function in SCILAB.

The code for constructing a filter is as follows:

ffilt('lp',20 , 0.2 , 0.5);

The following code calculates the frequency response of the filter.

#include <stdio.h>#include <stdlib.h>#include <math.h>#include <complex.h>double horner(double p[], int n, double x);double _Complex horner_C(double p[], int n, double _Complex x);int main(){    int i;    double p[] = {    -0.0196945,    -0.0356154,    1.559E-17,    0.0465740,    0.0340178,    -0.0415773,    -0.0864945,    1.559E-17,    0.2018205,    0.3741957,    0.3741957,    0.2018205,    1.559E-17,    -0.0864945,    -0.0415773,    0.0340178,    0.0465740,    1.559E-17,    -0.0356154,    -0.0196945    };    double x[201], mag[201], pha[201];    double _Complex xx;    for(i = 0; i <= 200; i++)    {        x[i] = 0.5 * i / 200.0;        xx = cexp(-2 * M_PI * x[i] * I);        mag[i] = cabs(horner_C(p, 20, xx));        pha[i] = carg(horner_C(p, 20, xx));        printf("%f, %f, %f\n", x[i] , mag[i], pha[i]);    }    return 0;}

The output results show the following amplitude-frequency characteristics:

The amplitude-frequency characteristics calculated by SCILAB are as follows:

We can see that the two images are exactly the same, indicating that our code calculation results are correct.