Scott Steffan NAE – Hmwk#10 12/2/99 10.2.3b Algorithm 10.1, Newton’s Method was used in Maple V to approximate solutions for x1 and x2 for the following nonlinear systems of equations within a 10-6 tolerance: y1 := ln( x 1 2x 2 2 )sin( x 1 x 2 )ln2ln( )0 y2 := e ( x 1x 2 ) cos( x 1 x 2 )0 Using an initial approximation of x(0)=(2,2)t the following was generated: > alg101(); This is the Newton Method for Nonlinear Systems. Input the number n of equations. >2 > ln(x1^2+x2^2)-sin(x1*x2)-1.83787706641 > exp(x1-x2)+cos(x1*x2) Input the Tolerance. > .000001 Input the maximum number of iterations. > 5000 Input initial approximation X(1). >2 Input initial approximation X(2). >2 Select output destination 1. Screen 2. Text file Enter 1 or 2 >1 Select amount of output 1. Answer only 2. All intermediate approximations Enter 1 or 2 >2 NEWTONS METHOD FOR NONLINEAR SYSTEMS Iteration, Approximation, Error 1 1.96868256 1.47890554 5.210945e-01 2 1.83008004 1.70902384 2.301183e-01 3 1.77555747 1.76841171 5.938786e-02 4 1.77246547 1.77243860 4.026893e-03 5 1.77245385 1.77245385 1.525135e-05 6 1.77245385 1.77245385 9.279643e-10 Iteration 6 gives solution: x(6) = (1.77245385, 1.77245385)t to within tolerance 1.0000000000e-06 After 6 iterations an approximation was within the specified accuracy. 10.3.2d Algorithm 10.2, Broyden’s Method was used in Maple V to approximate solutions to the nonlinear system shown below: y1 := x 1 2x 2370 y2 := x 1x 2 250 y3 := x 1x 2x 330 The tolerance was set to 10-6 and an initial approximation of x(0)=(0,0,0)t was used to generate the following: > alg102(); This is the Broyden’s Method for Nonlinear Systems. Input the number n of equations. >3 > x1^2+x2-37 > x1-x2^2-5 > x1+x2+x3-3 Input tolerance > 1e-6 Input the maximum number of iterations. > 50 Input initial approximation X(1). >0 Input initial approximation X(2). >0 Input initial approximation X(3). >0 Select output destination 1. Screen 2. Text file Enter 1 or 2 >1 Select amount of output 1. Answer only 2. All intermediate approximations Enter 1 or 2 >2 BROYDENS METHOD FOR NONLINEAR SYSTEMS Iteration, Approximation, 1 5.00000000 37.00000000 -39.00000000 Error 5.399074e+01 . . . Iteration number 38 gives solution: x(38) = (6.00000000, 1.00000000, -4.00000000)t to within tolerance 1.0000000000e-06 Process is complete! After 38 iterations an approximation was within the specified accuracy. 10.4.4a First, Algorithm 10.3, Steepest Descent was used to approximate solutions to the following nonlinear system to within 0.05: y1 := 15 x 1x 2 24 x 3130 y2 := x 1 210 x 2x 3110 y3 := x 2 325 x 3220 > alg103(); This is the Steepest Descent Method. Input the number n of equations. >3 Input the function CF[1](x1..x3). > 15*x1+x2^2-4*x3-13 Input the function CF[2](x1..x3). > x1^2+10*x2-x3-11 Input the function CF[3](x1..x3). > x2^3-25*x3+22 Input tolerance > .05 Input the maximum number of iterations. > 1000 Input initial approximation X(1). >0 Input initial approximation X(2). >0 Input initial approximation X(3). >0 Select output destination 1. Screen 2. Text file Enter 1 or 2 >1 Select amount of output 1. Answer only 2. All intermeditate approximations Enter 1 or 2 >2 STEEPEST DESCENT METHOD FOR NONLINEAR SYSTEMS Iteration 1 2 3 4 5 6 7 8 9 10 11 12 13 14 x1 approx 0.37709364 0.90159301 0.97977895 1.07563551 1.07603378 1.08541361 1.07236358 1.06980465 1.05955498 1.05702699 1.05049058 1.04888334 1.04488353 1.04393019 x2 approx 0.21271949 0.52037871 0.63117484 0.77475890 0.83410607 0.90767442 0.94041279 0.98077975 0.99960552 1.02296042 1.03412198 1.04794590 1.05467095 1.06292997 x3 approx 0.94176718 0.66238526 0.92257308 0.83161434 0.92559349 0.87925675 0.92827633 0.90072783 0.92925890 0.91288694 0.92986064 0.92014291 0.93033102 0.92450593 Iteration number 14 gives solution: x(14) = (1.04393019 1.06292997 .92450593)t to within 5.0000000000e-02 Once this was done, the results were used as initial approximations for a Newton’s Method approximation of the same nonlinear system of equations with a tighter tolerance of 10-6: > alg101(); This is the Newton Method for Nonlinear Systems. Input the number n of equations. >3 > 15*x1+x2^2-4*x3-13 > x1^2+10*x2-x3-11 > x2^3-25*x3+22 Input the Tolerance. > 1e-6 Input the maximum number of iterations. > 1000 Input initial approximation X(1). > 1.04393019 Input initial approximation X(2). > 1.06292997 Input initial approximation X(3). > .92450593 Select output destination 1. Screen 2. Text file Enter 1 or 2 >1 Select amount of output 1. Answer only 2. All intermediate approximations Enter 1 or 2 >2 NEWTONS METHOD FOR NONLINEAR SYSTEMS Iteration, Approximation, 1 1.03641778 1.08570188 . 93112417 2 1.03640047 1.08570655 . 93119144 3 1.03640047 1.08570655 . 93119144 Error 2.277191e-02 6.727593e-05 4.026260e-10 Iteration number 3 gives solution: x(3) = (1.03640047 1.08570655 .93119144)t to within tolerance 1.0000000000e-06 Using this technique allows Newton’s Method to predict very good approximations in much fewer iterations than choosing arbitrary initial approximations.