1. BRACKETING METHODS<br />Determine the real roots of f(x) = -0.5x2 + 2.5x + 4.5:<br />Graphically<br />Using the quadratic formula<br />Using three iterations of the bisection method to determine the highest root. Employ initial guesses of xt = 5 and xu = 10. Compute the estimated error Ɛa and the true error Ɛt after each iteration<br />Locate the first nontrivial root of sin x = x3, where x is in radians. Use a graphical technique and bisection with the initial interval from 0.5 to 1. Perform the computation until Ɛa is less Ɛs = 2%. Also perform an error check by substituting your final answer into the original equation.<br />A beam is loaded as shown in figure. Use the bisection method to solve for the position inside the beam where there is no moment.<br />100100lb/ft<br />4’3’2’3’<br />Determine the real root of f(x)=5x3-5x2+6x-2:<br />Graphically<br />Using bisection to locate the root. Employ initial guesses of xt=0 and xu=1 and iterate until the estimated error Ɛa falls below a level of Ɛs=10%.<br />Determine the positive real root of ln(x4)=0.7 (a) graphically, (b) using three iterations of the bisection method, with initial guesses of xl=0.5 and xu=2, and (c) using three iterations of the false-position method, with the same initial guesses as in (b).<br />Find the positive square root of 18 using the false-position method to within Ɛs=0.5%. Employ initial guesses of xl=4 and xu=5.<br />OPEN METHODS<br />Determine the real roots of f(x)=-1+5.5x-4x2+0.5x3: (a) graphically and (b) using the Newton-Raphson method to within Ɛs=0.01%.<br />Determine the roots of the following simultaneous nonlinear equations using (a) fixed-point iteration and (b) the Newton-Raphson method:<br />y=-x2+x+0.75<br />y+5xy=x2<br />employ initial guesses of x = y = 1.2 and discuss the results. <br />A mass balance for a pollutant in a well mixed lake can be written as<br />V dcdt=W-Qc-kVc<br />Given the parameter values V=1x106m3 /yr, W=1x106g/yr, and k=0.25m0.5/g0.5/yr, use the modified secant method to solve for the steady-state concentration. Employ an initial guess of c=4g/m3 and δ=0.5. Perform three iteration. <br />The “divide and average” method, an old-time method for approximating the square root of any positive number a, can be formulated as<br />x=x+a/x2<br />Prove that this is equivalent to the Newton-Raphson algorithm.<br />Locate the first positive root of<br />F(x) = sin x + cos (1+x2) – 1<br />Where x is in radians. Use four iterations of the secant method with initial guesses of (a) xi-1=1.0 and xi=3.0 (b) xi-1= 1.5 and xi=2.5 to locate the root. (d) Use the graphical method to explain your results.<br />Determine the real root of x3.5 = 80, with the modified secant method to within εs= 0.1% using an initial guess of xo= 3.5 and δ= 0.01<br />GAUSS ELIMINATION<br />Use Gauss elimination to solve:<br />8x1+2x2-2x3=-2<br />10x1+2x2+4x3=4<br />12x1+2x2+2x3=6<br />Employ partial pivoting and check your answers by substituting them into the original equations.<br />Given the system of equations<br />-3x2+7x3=2<br />X1+2x2-x3=3<br />5x1-2x2=2<br />Compute the determinant<br />Use Cramer’s rule to solve for the x’s.<br />Use Gauss elimination with partial pivoting to solve for the x’s. <br />Substitute your results back into the original equations to check your solution.<br />Use Gauss-Jordan elimination to solve:<br />2x1+x2-x3=-38<br />-5x1-x2+2x3=-34<br />3x1+x2-x3=-20<br />Do not employ pivoting. Check your answers by substituting them into the original equations.<br />Develop, debug, and test a program in either a high-level language or macro language of you choice to multiply two matrices-that is [X]=[Y][Z], where [Y] is m by n and [Z] is n by p.<br />LU DECOMPOSITION AND MATRIX INVERSION<br />Solve the following system of equations using LU decomposition with partial pivoting:<br />2x1 – 6x2 – x3 = - 38<br />-3x1 – x2 +7x3 = -34<br />-8x1 + x2 – 2x3 = -20<br />Determine the total flops as a function of the number of equations n for the (a) decomposition, (b) forward-substitution, and (c) back substitution phases of the LU decomposition version of Gauss elimination.<br />Use LU decomposition to determine the matrix inverse for the following system. Do not use a pivoting strategy, and check your results verifying that [A][A]-1=[I].<br />10X1 + 2X2 – X3 = 27<br />-3X1 – 6X2 + 2X3 = -61.5<br />X1 + X2 + 5X3 = -21.5<br />Use iterative refinement techniques to improve x1=2, x2=-3 and x3=8, which are approximate solutions of<br />2x1 + 5x2 + x3 = -5<br />6x1 + 2x2 + x3 = 12<br />x1 + 2x2 + x3 = 3<br />Let the function be defined on the interval [0,2] as follows:<br />F(x) =ax + b, 0<=x<=1<br />Cx + d, 1<=x<=2<br />Determine the constants a, b, c and d so that the function f satisfies the following:<br />F(0)=f(2)=1<br />F is continuous on the entire interval<br />a + b = 4<br />Source: <br />NUMERICAL METHODS FOR ENGINEERS<br />Fifth Edition<br />Steven C. Chapra<br />Raymond P. Canale<br />Mc Graw Hill<br />