Question

Selected values for the complementary error function are: * 0.2000 0.4000 0.8000 1.0000 1.4000 erfc(x) 0.7773 0.5716 0.2579 0.1573 0.0477 Estimate erfc(0.88), using (a) second-order, (b) third-order, (c) fourth-order, interpolating polynomials. Determine the percentage relative error in each case. Exact value can be obtained with the MATLAB built-in function erfc.

Answer 1

Solution: The value of table for x and y 0.2 0.4 0.8 1 1.4 y 0.7773 0.5716 0.2579 0.1573 0.0477 Langrange's Interpolating Polynomial Langrange's interpolation formula is (x-x1)(x - x2)(x-x3)(x-x4) (1 - x)(x - x)(x - 1)(x - x) (x - xo)(x-x1)(x - x3)(x-x4) (x - xa)(x-x1)(x - x2)(x- xa) (x-xo)(x-x1)(x-x2)(x- x3) f(x) = * yot *Y* * Y2+ *Y3 + * Y4 (xp-x1)(x - x2)(xo - x3)(xo-x4) (x1 - x0)(x1 - x2)(x1 - x3)(x -x4) (x2 - x)(x2-x1)(x2 - x3)(x2 - x4) (x3-xo)(x2-x1)(x3 - x2)(xy-x4) (x2-x)(x4-x1)(x2-x2)(x4- x3) (x -0.4)(x -0.8)(x - 1)(x - 1.4) (x -0.2)(x -0.8)(x - 1)(x - 1.4) (x -0.2)(x -0.4)(x - 1)(x - 1.4) (x -0.2)(x -0.4)(x -0.8)(x - 1.4) (x -0.2)(x -0.4)(x -0.8)(x- f(x) - (0.2 - 0.4)(0.2 - 0.8)(0.2 - 1)(0.2 - 1.4) * 0.7773+ (0.4 -0.2)(0.4 -0.8/0.4 - 180.4 - 1.4) * 0.5716 + (0.8 -0.2)(0.8 -0.4)(0.8 - 1)(0.8 -1.4) * 0.2579 + (1 -0.2)1 -0.4)(1 -0.8)(1 - 1.4) 0.1573 (1.4 -0.28(1.4 -0.4)(1.4 -0.8)(1. (x -0.4)(x -0.8)(x - 1)(x - 1.4) (x -0.2)(x -0.8)(x - 1)(x - 1.4) (x -0.2)(x -0.4)(x - 1)(x - 1.4) (x -0.2)(x -0.4)(x - 0.8)(x - 1.4) (x -0.2)(x -0.4)(x -0.8)(x - 1) f(x) = x 0.7773 + x 0.5716+ x 0.2579 + (-0.2)(-0.6)(-0.8)( - 1.2) * 0.1573 + (0.2)(-0.4)(-0.6)(-1) (0.6)(0.4)(-0.2) - 0.6) (0.8)(0.6)(0.2)(-0.4) (1.2)(1)(0.6)(0.4) x 0.0477 x4-3.6x3 + 4.6x2.2.45x +0.45 X4.3.4x3 + 3.96x2 - 1.78x + 0.22 x4.3x3 + 2.92x2.1.03x + 0.11 x4.2.8x3 +2.52x2 -0.85x + 0.09 x4-2.4x3 + 1.96x2 -0.62x + 0.06 f(x) = x 0.5716+ 0.7773 + 0.1152 x 0,0477 -0.048 * 0.2579 + 0.0288 x 0.1573 + -0.0384 0.288 fx) = (x4- 3.6x3 + 4.6x2 - 2.45x + 0.45) * 6.7474 + (x4- 3.4x3 + 3.96x2 - 1.78x + 0.22) * - 11.9083 + (x4- 3x3 + 2.92x2 - 1.03x +0.11) * 8.9549 + (x4-2.8x3 + 2.52x2 -0.85x + 0.09) * - 4.0964 + (x4-2.4x3 + 1.96x2 -0.62x + AX) = (6.75x4- 24.29x3 + 31.04x2- 16.52x + 3.02) + (-11.91x4 + 40.49x3- 47.16x2 + 21.24x - 2.67) + (8.95x4- 26.86x3 + 26.15x2- 9.24x + 1) + (-4.1x4 + 11.47x3- 10.32x2 + 3.47% - 0.37) + (0.17x4-0.4x3 + 0.32x2 - 0,18 +0 f(x) = -0.14x4 + 0.41x3 + 1.14x + 1