CE 5143 Homework 3 Due 10/20/07

— David Wagner 2007/09/20 16:23

Problem 1

f(x)=x^3-7x^2+12x-10=0, n=3, alpha=-1.994563, beta=1.992049,

a_3=1, a_2=-7, a_1=12, a_0=-10,

b_3=b_n=a_3=1;

b_2= b_{n-1} = a_{n-1} -alpha b_n =a_2 -alpha b_3 =-7+1.994563=-5.005437;

b_1=b_{n-2} = a_{n-2} -alpha b_{n-1} -beta b_n = a_1-alpha b_2 -beta b_3 = 12 -9.988365494 -1.992049 = 0.019585506;

b_0=b_{n-3} =a_0 -beta b_2 = -10+9.97107577=-0.02892423

{partial b_3}/{partial alpha} = 0; {partial b_2}/{partial alpha} = -b_3 =-1; {partial b_1}/{partial alpha} = -b_2 + alpha b_3 =5.005437-1.994563=3.010874; {partial b_0}/{partial alpha} = -beta {partial b_2}/{partial alpha}=1.992049

{partial b_3}/{partial beta}=0; {partial b_2}/{partial beta}=0; {partial b_1}/{partial beta}=-b_3=-1; {partial b_0}/{partial beta}=-b_2-beta{partial b_2}/{partial beta}=5.005437;

Delta alpha = { b_1{{partial b_0}/{partial beta}} - b_0{{partial b_1}/{partial beta}} } /{ {{partial b_0}/{partial alpha}}{{partial b_1}/{partial beta}} - {{partial b_1}/{partial alpha}}{{partial b_0}/{partial beta}} } = { (0.019585506)(5.005437) - (-0.02892423)(-1) } /{ (1.992049)(-1) - (3.010874)(5.005437) } = {0.069109786}/{-17.062789122} = -0.004050322

Delta beta= { b_0{{partial b_1}/{partial alpha}} - b_1{{partial b_0}/{partial alpha}} } /{ {{partial b_1}/{partial beta}}{{partial b_0}/{partial alpha}} - {{partial b_0}/{partial beta}}{{partial b_1}/{partial alpha}} } = { (-0.02892423)(3.010874) - (0.019585506)(1.992049) } /{ (-1)(1.992049) - (5.005437)(3.010874) } = {-0.087087212-0.039015288}/{-1.992049-15.070740122} = {-0.126102500}/{-17.062789122} = 0.007390497

alpha=alpha+Delta alpha=-1.994563-0.004050322=-1.998613322 beta=beta+Delta beta=1.992049+0.007390497=1.999439497

Δα and Δβ are small: overline{alpha}approx alpha, overline{beta} approx beta

x^2 +overline{alpha}x +overline{beta} =0; x={-overline{alpha}pm sqrt{overline{alpha}^2 -4 overline{beta}}}/2 = {-(-1.998613322)pm sqrt{(-1.998613322)^2 -4 (1.999439497)}}/2 ={1.998613322 pm sqrt{-4.00330277}}/2 ={1.998613322 pm 2.000825524i}/2;

x_1=0.999307+1.000413i, x_2= 0.999307-1.000413i

The remaining polynomial coefficients are: a_1=b_3=,a_0=b_2=-5.005437, or x-5.005437=0 → x_3=5.005437

0.999307+1.000413i, 0.999307-1.000413i, 5.005437

Problem 2

f_1(x,y)=cosh x-y=0 and f_2(x,y)=xy-1=0, epsilon=0.001

i=1; delim{[}{matrix{2}{2}{{{partial f_1}/{partial x}} {{partial f_1}/{partial y}} {{partial f_2}/{partial x}} {{partial f_2}/{partial y}}}}{]} delim{lbrace}{matrix{2}{1}{{Delta x} {Delta y}}}{rbrace} = delim{lbrace}{matrix{2}{1}{{-f_1} {-f_2}}}{rbrace} =delim{[}{matrix{2}{2}{{sinhx} {-1} {1} {1}} delim{lbrace}{matrix{2}{1}{{Delta x} {Delta y}}}{rbrace}}{]} = delim{lbrace}{matrix{2}{1}{{y-cosh x} {1-xy}}}{rbrace}

start with (1.0,1.0)

delim{[}{matrix{2}{2}{{1.175201} {-1} {1} {1}}}{]} delim{lbrace}{matrix{2}{1}{{Delta x} {Delta y} }}{rbrace} = delim{lbrace}{matrix{2}{1}{{-0.543081} {0}}}{rbrace}

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