Polynomial functions and analysis.

Functions
int DSPL_API	cheby_poly1 (double *x, int n, int ord, double *y)
	Chebyshev polynomial of the first kind order ord More...
int DSPL_API	cheby_poly2 (double *x, int n, int ord, double *y)
	Chebyshev polynomial of the second kind order ord More...
int DSPL_API	polyroots (double *a, int ord, complex_t *r, int *info)
	Function calculates real polynomial roots. More...

Detailed Description

Function Documentation

cheby_poly1()

int cheby_poly1	(	double *	x,
		int	n,
		int	ord,
		double *	y
	)

Chebyshev polynomial of the first kind order ord

---

Function calculates Chebyshev polynomial \( C_{ord}(x)\) of the first kind order ord for the real vector x (length n) by recurrent equation:

\[ C_{ord}(x) = 2 x C_{ord-1}(x) - C_{ord-2}(x), \]

where \( C_0(x) = 1 \), \( C_1(x) = x\)

Parameters

[in]	x	Pointer to the real argument vector x. Vector size is [n x 1].
[in]	n	Size of vectors x and y.
[in]	ord	Chebyshev polynomial order.
[out]	y	Pointer to the Chebyshev polynomial values, corresponds to the argument x. Vector size is [n x 1]. Memory must be allocated.

Returns

RES_OK if Chebyshev polynomial is calculated successfully.
Else code error.
Example:

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include "dspl.h"
 
#define N 250
 
int main(int argc, char* argv[])
{
    void* hdspl;  /* DSPL handle        */
    void* hplot;  /* GNUPLOT handle     */
    double x[N], y[N];
    int ord;
    char fn[64];
 
 
    hdspl = dspl_load();   /* Load DSPL function */
 
    linspace(-1.0, 1.0, N, DSPL_SYMMETRIC, x);
    for(ord = 1; ord < 5; ord++)
    {
        cheby_poly1(x, N, ord, y);
        sprintf(fn, "dat/cheby_poly1_ord%d.txt", ord);
        writetxt(x,y,N,fn);
    }
 
    /* plotting by GNUPLOT */
    gnuplot_create(argc, argv, 560, 380, "img/cheby_poly1.png", &hplot);
    gnuplot_cmd(hplot, "set grid");
    gnuplot_cmd(hplot, "set key left top");
    gnuplot_cmd(hplot, "set xlabel 'x'");
    gnuplot_cmd(hplot, "set ylabel 'C_N(x)'");
    gnuplot_cmd(hplot, "set yrange [-1.5:1.5]");
    gnuplot_cmd(hplot, "plot 'dat/cheby_poly1_ord1.txt' with lines, \\");
    gnuplot_cmd(hplot, "     'dat/cheby_poly1_ord2.txt' with lines, \\");
    gnuplot_cmd(hplot, "     'dat/cheby_poly1_ord3.txt' with lines, \\");
    gnuplot_cmd(hplot, "     'dat/cheby_poly1_ord4.txt' with lines");
    gnuplot_close(hplot);
 
    dspl_free(hdspl);      /* free dspl handle */
 
    return 0;
}

Text files will be created in dat directory:

cheby_poly1_ord1.txt
cheby_poly1_ord2.txt
cheby_poly1_ord3.txt
cheby_poly1_ord4.txt

GNUPLOT package will create Chebyshev polynomials plot from saved text-files:

Author

Sergey Bakhurin www.dsplib.org

Definition at line 136 of file cheby_poly1.c.

cheby_poly2()

int cheby_poly2	(	double *	x,
		int	n,
		int	ord,
		double *	y
	)

Chebyshev polynomial of the second kind order ord

---

Function calculates Chebyshev polynomial \( U_ord(x)\) of the first kind order ord for the real vector x (length n) by recurrent equation:

\[ U_{ord}(x) = 2 x U_{ord-1}(x) - U_{ord-2}(x), \]

where \( U_0(x) = 1 \), \( U_1(x) = 2x\)

Parameters

[in]	x	Pointer to the real argument vector x. Vector size is [n x 1].
[in]	n	Size of vectors x and y.
[in]	ord	Chebyshev polynomial order.
[out]	y	Pointer to the Chebyshev polynomial values, corresponds to the argument x. Vector size is [n x 1]. Memory must be allocated.

Returns

RES_OK if Chebyshev polynomial is calculated successfully.
Else code error.
Example:

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include "dspl.h"
 
#define N 250
 
int main(int argc, char* argv[])
{
    void* hdspl;  /* DSPL handle        */
    void* hplot;  /* GNUPLOT handle     */
 
 
    double x[N], y[N];
    int ord;
    char fn[64];
    hdspl = dspl_load();   /* Load DSPL function */
    linspace(-1.0, 1.0, N, DSPL_SYMMETRIC, x);
    for(ord = 1; ord < 5; ord++)
    {
        cheby_poly2(x, N, ord, y);
        sprintf(fn, "dat/cheby_poly2_ord%d.txt", ord);
        writetxt(x,y,N,fn);
    }
 
    /* plotting by GNUPLOT */
    gnuplot_create(argc, argv, 560, 380, "img/cheby_poly2.png", &hplot);
    gnuplot_cmd(hplot, "set grid");
    gnuplot_cmd(hplot, "set key left top");
    gnuplot_cmd(hplot, "set xlabel 'x'");
    gnuplot_cmd(hplot, "set ylabel 'U_N (x)'");
    gnuplot_cmd(hplot, "set yrange [-3.5:3.5]");
    gnuplot_cmd(hplot, "plot 'dat/cheby_poly2_ord1.txt' with lines, \\");
    gnuplot_cmd(hplot, "     'dat/cheby_poly2_ord2.txt' with lines, \\");
    gnuplot_cmd(hplot, "     'dat/cheby_poly2_ord3.txt' with lines, \\");
    gnuplot_cmd(hplot, "     'dat/cheby_poly2_ord4.txt' with lines");
    gnuplot_close(hplot);
 
    dspl_free(hdspl);      /* free dspl handle */
 
    return 0;
}

Text files will be created in dat directory:

cheby_poly2_ord1.txt
cheby_poly2_ord2.txt
cheby_poly2_ord3.txt
cheby_poly2_ord4.txt

GNUPLOT package will create Chebyshev polynomials plot from saved text-files:

Author

Sergey Bakhurin www.dsplib.org

Definition at line 133 of file cheby_poly2.c.

polyroots()

int polyroots	(	double *	a,
		int	ord,
		complex_t *	r,
		int *	info
	)

Function calculates real polynomial roots.

---

Function calculates roots of the real polynomial \(P_N(x)\) order \(N\) with a coefficient vector size [(N+1) x 1].

\[ P_N(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ... a_N x^N. \]

The roots of the polynomial are calculated as eigenvalues of the polynomial companion matrix. To calculate the eigenvalues, a subroutine of the LAPACK package is used.

Parameters

[in]	a	Pointer to the vector of coefficients. Vector size is [ord+1 x 1]. Coefficient a[0] corresponds to the \(a_0\) polynomial coefficient. Coefficient a[ord] cannot be zero.
[in]	ord	Polynomial order \(N\).
[out]	r	Pointer to the polynomial roots vector. Vector size is [ord x 1]. Memory must be allocated. The roots of a real polynomial can be either real or form simple or multiple complex conjugate pairs of roots. Therefore, the output root vector is of a complex data type.
[out]	info	Pointer to the LAPACK subroutine error code. This code is returned by the LAPACK subroutine and translated through this variable for analysis..

Returns

RES_OK — roots are calculated successfully.

Else code error.

Example:

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include "dspl.h"
 
#define N  2
 
int main(int argc, char* argv[])
{
    void* hdspl;                     /* DSPL handle           */
    double a[N+1] = {2.0, 2.0, 1.0}; /* P(x) = 2 + 2x + x^2   */
    complex_t r[N] = {0};            /* roots                 */
    int err, n, info;
    hdspl = dspl_load();            /* Load DSPL functions    */
    if(!hdspl)
    {
        printf("libdspl loading error!\n");
        return -1;
    }
 
 
    /* roots calculation */
    err = polyroots(a, N, r, &info);
    printf("Error code: 0x%.8x\n", err);
 
    /* print roots */
    for(n = 0; n < N; n++)
        printf("r[%d] = % -8.5f% -8.5f j\n", n, RE(r[n]), IM(r[n]));
 
    /* free dspl handle  */
    dspl_free(hdspl);
    return 0;
}
 

This program calculates the roots of the polynomial

\[ P(x) = 2 + 2x + x^2 \]

and prints the calculated roots. The result of the program:

Error code: 0x00000000
r[0] = -1.00000 1.00000 j
r[1] = -1.00000-1.00000 j

Author

Sergey Bakhurin. www.dsplib.org

Definition at line 157 of file polyroots.c.