************* * EXAMPLE 1 * ************* f(x) = cos(x) - x x within 0 and pi/2 ----------------- >> newton Input the function f(x) in terms of x cos(x) - x Input the derivative of function f(x) in terms of x -sin(x) -1 Enter tolerance) 0.00001 Enter maximum number of iterations) 100 Enter initial guess) 0.6 iteration number 1 p is 7.44017319e-01 iteration number 2 p is 7.39090477e-01 iteration number 3 p is 7.39085133e-01 result after iteration number 3 is 7.39085133e-01 >> bisection Enter tol, a, b, nmax (on separate lines) 0.00001 0.4 0.8 100 Input the function f(x) in terms of x cos(x) - x result after iteration number 16 is 7.39080811e-01 >> bisection Enter tol, a, b, nmax (on separate lines) 0.00001 0.73 0.74 100 Input the function f(x) in terms of x cos(x) - x result after iteration number 11 is 7.39086914e-01 ************* * EXAMPLE 2 * ************* f(x) = x^3 + 3*x^2 - 1 x within [-3,-2] ------------------------------------------ >> newton Input the function f(x) in terms of x x^3 + 3*x^2 - 1 Input the derivative of function f(x) in terms of x 3*x^2 + 6*x Enter tolerance) 0.00001 Enter maximum number of iterations) 100 Enter initial guess) -2.5 iteration number 1 p is -3.06666667e+00 iteration number 2 p is -2.90087560e+00 iteration number 3 p is -2.87971990e+00 iteration number 4 p is -2.87938532e+00 iteration number 5 p is -2.87938524e+00 result after iteration number 5 is -2.87938524e+00 >> bisection Enter tol, a, b, nmax (on separate lines) 0.00001 -3 -2 100 Input the function f(x) in terms of x x^3 + 3*x^2 - 1 result after iteration number 16 is -2.87937927e+00 >> bisection Enter tol, a, b, nmax (on separate lines) 0.00001 -2.9 -2.8 100 Input the function f(x) in terms of x x^3 + 3*x^2 - 1 result after iteration number 12 is -2.87937012e+00 **************************************** * EXAMPLE 3: Lennard - Jones Potential * **************************************** f(x) = x^-12 - x^-6 x between 0 and infinity ------------------------------------------ >> bisection Enter tol, a, b, nmax (on separate lines) 0.0001 0.1 3 1000 Input the function f(x) in terms of x x^-12 - x^-6 result after iteration number 15 is 9.99966431e-01 >> newton Input the function f(x) in terms of x x^-12 - x^-6 Input the derivative of function f(x) in terms of x -12*x^-13 + 6*x^-7 Enter tolerance) 0.0001 Enter maximum number of iterations) 1000 Enter initial guess) 0.9 iteration number 1 p is 9.47859058e-01 iteration number 2 p is 9.81912206e-01 iteration number 3 p is 9.97293246e-01 iteration number 4 p is 9.99931908e-01 iteration number 5 p is 9.99999956e-01 result after iteration number 5 is 9.99999956e-01 >> newton Input the function f(x) in terms of x x^-12 - x^-6 Input the derivative of function f(x) in terms of x -12*x^-13 + 6*x^-7 Enter tolerance) 0.0001 Enter maximum number of iterations) 20 Enter initial guess) 1.1 iteration number 1 p is 4.80785024e-01 iteration number 2 p is 5.20601479e-01 iteration number 3 p is 5.63548750e-01 iteration number 4 p is 6.09746745e-01 iteration number 5 p is 6.59218862e-01 iteration number 6 p is 7.11803090e-01 iteration number 7 p is 7.66994185e-01 iteration number 8 p is 8.23666719e-01 iteration number 9 p is 8.79606525e-01 iteration number 10 p is 9.30816389e-01 iteration number 11 p is 9.71002333e-01 iteration number 12 p is 9.93546394e-01 iteration number 13 p is 9.99624208e-01 iteration number 14 p is 9.99998663e-01 iteration number 15 p is 1.00000000e+00 result after iteration number 15 is 1.00000000e+00 >> newton Input the function f(x) in terms of x x^-12 - x^-6 Input the derivative of function f(x) in terms of x -12*x^-13 + 6*x^-7 Enter tolerance) 0.0001 Enter maximum number of iterations) 20 Enter initial guess) 2 iteration number 1 p is 2.33870968e+00 iteration number 2 p is 2.73090624e+00 iteration number 3 p is 3.18715987e+00 iteration number 4 p is 3.71886094e+00 iteration number 5 p is 4.33890559e+00 iteration number 6 p is 5.06216493e+00 iteration number 7 p is 5.90590923e+00 iteration number 8 p is 6.89025064e+00 iteration number 9 p is 8.03863647e+00 iteration number 10 p is 9.37841419e+00 iteration number 11 p is 1.09414855e+01 iteration number 12 p is 1.27650675e+01 iteration number 13 p is 1.48925792e+01 iteration number 14 p is 1.73746760e+01 iteration number 15 p is 2.02704554e+01 iteration number 16 p is 2.36488647e+01 iteration number 17 p is 2.75903422e+01 iteration number 18 p is 3.21887326e+01 iteration number 19 p is 3.75535214e+01 iteration number 20 p is 4.38124416e+01 iteration number 21 p is 5.11145152e+01 no convergence after iteration number 20