Linear Algebra and the C Language/a07j

__NOTOC__ Install and compile this file in your working directory. /* ------------------------------------ */ /* Save as : c00c.c */ /* ------------------------------------ */ #include "v_a.h" #include "dpoly.h" /* ------------------------------------ */ int main(void) { double xy[8] ={ -5, -3, -2, 0, 2, 3, 3, -2 }; double **XY = ca_A_mR(xy,i_mR(R4,C2)); double **A = i_mR(R4,C4); double **b = i_mR(R4,C1); double **Ab = i_Abr_Ac_bc_mR(R4,C4,C1); clrscrn(); printf("\n"); printf(" Find the coefficients a, b, c of the curve \n\n"); printf(" y = ax**3 + bx**2 + cx + d \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**3 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]); verify_X_mR(Ab,XY[R4][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**3 + bx**2 + cx + d Which passes through these four points. x[1], y[1] x[2], y[2] x[3], y[3] x[4], y[4] Using the points we obtain the matrix: x**3 x**2 x**1 x**0 y x[1]**3 x[1]**2 x[1]**1 x[1]**0 y[1] x[2]**3 x[2]**2 x[2]**1 x[2]**0 y[2] x[3]**3 x[3]**2 x[3]**1 x[3]**0 y[3] x[4]**3 x[4]**2 x[4]**1 x[4]**0 y[4] That we can write: x**3 x**2 x 1 y x[1]**3 x[1]**2 x[1] 1 y[1] x[2]**3 x[2]**2 x[2] 1 y[2] x[3]**3 x[3]**2 x[3] 1 y[3] x[4]**3 x[4]**2 x[4] 1 y[4] Let's use the gj_TP_mR() function to solve the system that will give us the coefficients a, b, c, d Screen output example: Find the coefficients a, b, c of the curve y = ax**3 + bx**2 + cx + d that passes through the points. x y -5 -3 -2 +0 +2 +3 +3 -2 Using the given points, we obtain this matrix x**3 x**2 x**1 x**0 y -125.00 +25.00 -5.00 +1.00 -3.00 -8.00 +4.00 -2.00 +1.00 +0.00 +8.00 +4.00 +2.00 +1.00 +3.00 +27.00 +9.00 +3.00 +1.00 -2.00 Press return to continue. The Gauss Jordan process will reduce this matrix to : +1.00 +0.00 +0.00 +0.00 -0.14 +0.00 +1.00 +0.00 +0.00 -0.73 +0.00 +0.00 +1.00 +0.00 +1.31 +0.00 +0.00 +0.00 +1.00 +4.43 The coefficients a, b, c of the curve are : y = -0.139x**3 -0.732x**2 +1.307x +4.429 Press return to continue. x y -5 -3 -2 +0 +2 +3 +3 -2 Verify the result : With x = -5.000, y = -3.000 With x = -2.000, y = +0.000 With x = +2.000, y = +3.000 With x = +3.000, y = -2.000 Press return to continue. Copy and paste in Octave: function y = f (x) y = -0.139285714*x^3 -0.732142857*x^2 +1.307142857*x +4.428571429; endfunction f (-5) f (-2) f (+2) f (+3)