411 REFERENCE · Wikibooks
Linear Algebra and the C Language/a07h
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__NOTOC__
Install and compile this file in your working directory.
/* ------------------------------------ */
/* Save as : c00a.c */
/* ------------------------------------ */
#include "v_a.h"
#include "dpoly.h"
/* ------------------------------------ */
int main(void)
{
double xy[6] ={1, 6,
2, 3,
3, 5 };
double **XY = ca_A_mR(xy,i_mR(R3,C2));
double **A = i_mR(R3,C3);
double **b = i_mR(R3,C1);
double **Ab = i_Abr_Ac_bc_mR(R3,C3,C1);
clrscrn();
printf("\n");
printf(" Find the coefficients a, b, c of the curve \n\n");
printf(" y = ax**2 + bx + c (x**0 = 1) \n\n");
printf(" that passes through the points. \n\n");
printf(" x y \n");
p_mR(XY,S5,P0,C6);
printf("\n Using the given points, we obtain this matrix\n\n");
printf(" x**2 x**1 x**0 y\n");
i_A_b_with_XY_mR(XY,A,b);
c_A_b_Ab_mR(A,b,Ab);
p_mR(Ab,S7,P2,C6);
stop();
clrscrn();
printf(" The Gauss Jordan process will reduce this matrix to : \n");
gj_TP_mR(Ab);
p_mR(Ab,S7,P2,C6);
printf("\n The coefficients a, b, c of the curve are : \n\n");
p_eq_poly_mR(Ab);
stop();
clrscrn();
printf(" x y \n");
p_mR(XY,S5,P0,C6);
printf("\n");
printf(" Verify the result : \n\n");
verify_X_mR(Ab,XY[R1][C1]);
verify_X_mR(Ab,XY[R2][C1]);
verify_X_mR(Ab,XY[R3][C1]);
printf("\n\n\n");
stop();
f_mR(XY);
f_mR(A);
f_mR(b);
f_mR(Ab);
return 0;
}
/* ------------------------------------ */
/* ------------------------------------ */
Presentation :
Let's calculate the coefficients of a polynomial.
y = ax**2 + bx + c
Which passes through these three points.
x[1], y[1]
x[2], y[2]
x[3], y[3]
Using the points we obtain the matrix:
x**2 x**1 x**0 y
x[1]**2 x[1]**1 x[1]**0 y[1]
x[2]**2 x[2]**1 x[2]**0 y[2]
x[3]**2 x[3]**1 x[3]**0 y[3]
That we can write:
x**2 x 1 y
x[1]**2 x[1] 1 y[1]
x[2]**2 x[2] 1 y[2]
x[3]**2 x[3] 1 y[3]
Let's use the gj_TP_mR() function to solve
the system that will give us the coefficients a, b, c
Screen output example:
Find the coefficients a, b, c of the curve
y = ax**2 + bx + c (x**0 = 1)
that passes through the points.
x y
+1 +6
+2 +3
+3 +5
Using the given points, we obtain this matrix
x**2 x**1 x**0 y
+1.00 +1.00 +1.00 +6.00
+4.00 +2.00 +1.00 +3.00
+9.00 +3.00 +1.00 +5.00
Press return to continue.
The Gauss Jordan process will reduce this matrix to :
+1.00 +0.00 +0.00 +2.50
+0.00 +1.00 +0.00 -10.50
+0.00 +0.00 +1.00 +14.00
The coefficients a, b, c of the curve are :
y = +2.500x**2 -10.500x +14.000
Press return to continue.
x y
+1 +6
+2 +3
+3 +5
Verify the result :
With x = +1.000, y = +6.000
With x = +2.000, y = +3.000
With x = +3.000, y = +5.000
Press return to continue.
Copy and paste in Octave:
function y = f (x)
y = +2.50*x^2 -10.50*x +14.00;
endfunction
f (+1)
f (+2)
f (+3)