Download LU Factorization using Crout's Method: Homework Assignment Solution Set and more Assignments Civil Engineering in PDF only on Docsity! Homework Assignment 10 – Solution Set Problem 1 Problem a) f1(x,y,z) = x3 - 10x + y + z + 3 = 0 f2(x,y,z) = y3 + 10y - 2x - 2z - 5 = 0 f3(x,y,z) = x + y - 10z + 2sin(z) + 5 =0 The scalar function is h(x,y,z) = (x3 - 10x + y + z + 3)2 + (y3 + 10y - 2x - 2z - 5)2 + ( x + y - 10z + 2sin(z) + 5 )2 dh/dx = 2*(x3 - 10x + y + z + 3)*(3x2-10) +2*(y3 + 10y - 2x - 2z - 5)*(-2) +2*( x + y - 10z +2sin(z)+5)*(1) dh/dy = 2*(x3 - 10x + y + z + 3)*(1) +2*(y3 + 10y - 2x - 2z - 5)*( 3y2+10) +2*( x + y - 10z + 2sin(z)+5)*(1) dh/dz = 2*(x3-10x+y+z+3)*(1) + 2*(y3+10y-2x-2z-5)*(-2) + 2*( x + y - 10z + 2sin(z) + 5)*(2cos(z)-10) Test 1 x0 = {1,1,1}T >> FFMIN_HW10a(x0,tol,30) k x(1) x(2) x(3) norm(x) 0 1.00000 1.00000 1.00000 0.00e+000 1 0.29116 0.35366 0.88289 -3.03e+000 2 0.34786 0.66128 0.70170 -1.77e+001 and so on … 8 0.45501 0.70677 0.75241 -9.92e-005 9 0.45531 0.70618 0.75277 -6.00e-005 iterations converge Test 2 x0 = {-1,-1,-1}T FFMIN_HW10a(x1,tol,30) k x(1) x(2) x(3) norm(x) 0 -1.00000 -1.00000 -1.00000 0.00e+000 1 -0.45808 1.10442 0.04596 -2.02e+002 2 0.26120 0.11901 0.43687 -1.38e+002 and so on … 10 0.45468 0.70547 0.75194 -4.25e-004 11 0.45502 0.70662 0.75255 -8.82e-005 12 0.45529 0.70622 0.75280 -3.12e-005 iterations converge Test 3 x0 = {-1,2,5}T FFMIN_HW10a(x2,tol,30) k x(1) x(2) x(3) norm(x) 0 -1.00000 2.00000 5.00000 0.00e+000 1 0.63217 -0.36783 1.70212 -2.42e+003 2 0.37993 0.81118 0.72275 -2.75e+002 and so on … 7 0.45523 0.70559 0.75256 -9.64e-004 8 0.45509 0.70632 0.75258 -6.30e-005 iterations converge Problem b f1(x,y,z) = 10x - x2 - y - z2 - 4 = 0 f2(x,y,z) = 10y - x2 - y2 – z - 5 = 0 f3(x,y,z) = 10z - x - y2 - z2 - 6 = 0 The scalar function h(x,y,z) = (10x - x2 - y - z2 - 4 )2 + (10y - x2 - y2 – z - 5 )2 + (10z - x - y2 - z2 - 6 )2 dh/dx =2*(10x - x2 - y - z2 - 4 )*(10-2x) + 2*(10y - x2-y2 – z - 5)*(-2x) + 2*(10z - x-y2- z2-6 )*(-1) dh/dy =2*(10x - x2 - y - z2 - 4 )*(-1) + 2*(10y - x2-y2 – z - 5)*(10-2y) + 2*(10z - x-y2- z2-6 )*(-2y) dh/dz =2*(10x - x2 - y - z2 - 4 )*(-2z) + 2*(10y - x2-y2 – z - 5)*(-1) + 2*(10z - x-y2- z2-6 )*(10-2z) Test 1 x0 = {1,1,1}T FFMIN_HW10b(x0,tol,30) k x(1) x(2) x(3) norm(x) 0 1.00000 1.00000 1.00000 0.00e+000 1 0.40625 0.65625 1.00000 -6.23e+000 and so on … 11 0.55158 0.64794 0.75447 -1.10e-004 12 0.55244 0.64782 0.75381 -3.38e-005 13 0.55196 0.64789 0.75417 -1.04e-005 iterations converge Test 2 x0 = {-1,-1,-1}T FFMIN_HW10b(x1,tol,30) k x(1) x(2) x(3) norm(x) 0 -1.00000 -1.00000 -1.00000 0.00e+000 1 2.04688 2.29688 2.40625 -6.91e+002 2 2.20783 2.10838 2.42481 -5.55e-001 and so on … 27 0.55156 0.64801 0.75444 -1.13e-004 28 0.55245 0.64779 0.75383 -3.47e-005 29 0.55195 0.64790 0.75417 -1.07e-005 iterations converge Test 3 x0 = {-1,2,5}T FFMIN_HW10b(x2,tol,30) k x(1) x(2) x(3) norm(x) 0 -1.00000 2.00000 5.00000 0.00e+000 1 2.98438 1.93750 1.75781 -1.90e+003 2 1.53064 2.52986 2.71032 -3.19e+001 and so on … 16 3.80644 3.76194 3.97778 -1.07e-003 17 3.80678 3.76136 3.97675 -8.93e-005 iterations converge
