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ROOTS OF EQUATIONS Student Notes ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier Dr. B.A. DeVantier Applied Problem The concentration of pollutant bacteria C in a lake decreases according to: C 80e2 t 20e0.1t Determine the time required for the bacteria to be reduced to 10 ppm. Applied Problem You buy a $20 K piece of equipment for nothing down and $5K per year for 5 years. What interest rate are you paying? The formula relating present worth (P), annual payments (A), number of years (n) and the interest rate (i) is: i1 i A P n 1 i 1 n Quadratic Formula b b 2 4 ac x 2a f ( x) ax 2 bx c 0 This equation gives us the roots of the algebraic function f(x) i.e. the value of x that makes f(x) = 0 How can we solve for f(x) = e-x - x? Roots of Equations Plot the function and determine where it crosses the x-axis Lacks precision Trial and error f(x) 10 8 6 4 2 0 -2 -2 -1 0 x 1 2 Overview of Methods Bracketing methods Bisection method False position Open methods Newton-Raphson Secant method Specific Study Objectives Understand the graphical interpretation of a root Know the graphical interpretation of the false-position method (regula falsi method) and why it is usually superior to the bisection method Understand the difference between bracketing and open methods for root location Specific Study Objectives Understand the concepts of convergence and divergence. Know why bracketing methods always converge, whereas open methods may sometimes diverge Know the fundamental difference between the false position and secant methods and how it relates to convergence Specific Study Objectives Understand the problems posed by multiple roots and the modification available to mitigate them Use the techniques presented to find the root of an equation Solve two nonlinear simultaneous equations using techniques similar to root finding methods Bracketing Methods Bisection method False position method (regula falsi method) Graphically Speaking 1. 2. 3. 4. 5. 6. 7. Graph the function Based on the graph, select two x values that “bracket the root” What is the sign of the y value? Determine a new x (xr) based on the method What is the sign of the y value of xr? Switch xr with the point that has a y value with the same sign Continue until f(xr) = 0 xl xr xu Theory Behind Bracketing Methods f(x) f(x) x consider lower and upper bound same sign, no roots or even # of roots x f(x) f(x) opposite sign, odd # of roots x x Bisection Method xr = (xl + xu)/2 Takes advantage of sign changing There is at least one real root f(x) x Graphically Speaking 1. 2. 3. 4. 5. 6. 7. Graph the function Based on the graph, select two x values that “bracket the root” What is the sign of the y value? xr = (xl + xu)/2 What is the sign of the y value of xr? Switch xr with the point that has a y value with the same sign Continue until f(xr) = 0 xl xr xu Algorithm Choose xu and xl. Verify sign change f(xl)f(xu) < 0 Estimate root xr = (xl + xu) / 2 Determine if the estimate is in the lower or upper subinterval f(xl)f(xr) < 0 then xu = xr RETURN f(xl)f(xr) >0 then xl = xr RETURN f(xl)f(xr) =0 then root equals xr - COMPLETE Error present approx previous approx a 100 present approx Let’s consider an example problem: Example Use the bisection method to determine the root -x xu = 1 10 8 6 f(x) = 3.718 f(x) •f(x) = •xl = -1 e-x 4 2 0 -2 -1.5 -1 -0.5 0 -2 -4 0.5 1 1.5 f(x) = -0.632 x STRATEGY 2 Strategy Calculate f(xl) and f(xu) Calculate xr Calculate f(xr) Replace xl or xu with xr based on the sign of f(xr) Calculate a based on xr for all iterations after the first iteration REPEAT False Position Method “Brute Force” of bisection method is inefficient Join points by a straight line Improves the estimate Replacing the curve by a straight line gives the “false position” next estimate, xr f(xu) xl f(xl) xu Based on similar triangles f xl f xu xr xl xr xu f xu xl xu xr xu f xl f xu Example Determine the root of the following equation using the false position method starting with an initial estimate of xl=4.55 and xu=4.65 30 f(x) = x3 - 98 20 f(x) 10 0 -1 0 -20 -30 -40 4 4.5 5 x STRATEGY Strategy Calculate f(xl) and f(xu) Calculate xr Calculate f(xr) Replace xl or xu with xr based on the sign of f(xr) Calculate a based on xr for all iterations after the first iteration REPEAT Example Spreadsheet Use of IF-THEN statements Recall in the bi-section or false position methods. If f(xl)f(xr)>0 then they are the same sign Need to replace xu with xr If f(xl)f(xr)< 0 then they are opposite signs Need to replace xl with xr Example Spreadsheet xl xu f(xl) f(xu) xr f(xr) 0.01 0.10 -549.03 592.15 0.06 3.58 f(xl)f(xr) -1964.96 ? If f(xl)f(xr) is negative, we want to leave xu as xu If f(xl)f(xr) is positive, we want to replace xu with xr The EXCEL command for the next xu entry follows the logic If f(xl)f(xr) < 0, xu,xr Example Spreadsheet Pitfalls of False Position Method f(x) f(x)=x1 0-1 30 25 20 15 10 5 0 -5 0 0.5 1 x 1 .5 Open Methods Newton-Raphson method Secant method Multiple roots In the previous bracketing methods, the root is located within an interval prescribed by an upper and lower boundary Newton Raphson most widely used f(x) x Newton Raphson tangent f(xi) dy tangent f' dx f xi 0 f ' xi xi xi1 xi+1 xi rearrange f xi xi1 xi f ' xi Newton Raphson xi 1 xi i 0 1 f xi f ' xi x A D f(x) B f’(x) C 2 A is the initial estimate B is the function evaluated at A C is the first derivative evaluated at A D= A-B/C Repeat Newton Raphson Pitfalls Solution can “overshoot” the root and potentially diverge f(x) x2 x1 x0 x Use the Newton Raphson method to determine the root of f(x) = x2 - 11 using an initial guess of xi = 3 6 f(x) Example 4 2 0 -2 x 0 1 2 3 4 -4 -6 -8 -10 -12 STRATEGY 5 Strategy Start a table to track your solution i xi 0 x0 f(xi) f’(xi) Calculate f(x) and f’(x) Estimate the next xi based on the governing equation Use s to determine when to stop Note: use of subscript “0” Secant method Approximate derivative using a finite divided difference f xi 1 f xi f ' x xi 1 xi What is this? HINT: dy / dx = Dy / Dx Substitute this into the formula for Newton Raphson f xi xi1 xi f ' xi f xi xi 1 xi xi1 xi f xi 1 f xi Substitute finite difference approximation for the first derivative into this equation for Newton Raphson Secant method Secant method f xi xi1 xi xi1 xi f xi1 f xi Requires two initial estimates f(x) is not required to change signs, therefore this is not a bracketing method Secant method f(x) { slope between two estimates new estimate x initial estimates Example Determine the root of f(x) = e-x - x using the secant method. Use the starting points x0 = 0 and x1 = 1.0. 1.5 0, 1.000 1.0 f(x) 0.5 0.0 -0.5 -1.0 0 0.5 1 1.5 2 2.5 x 1, -0.632 -1.5 -2.0 -2.5 STRATEGY Strategy Start a table to track your results i xi f(xi) 0 0 Calculate 1 1 Calculate 2 Calculate a Note: here you need two starting points! Use these to calculate x2 Use x3 and x2 to calculate a at i=3 Use s Comparison of False Position and Secant Method 2 f(x) f(x) 2 1 x 1 new est. x new est. Multiple Roots Corresponds to a point where a function is tangential to the x-axis i.e. double root 5x2 f(x) = + 7x -3 f(x) = (x-3)(x-1)(x-1) i.e. triple root f(x) = (x-3)(x-1)3 8 6 f(x) x3 10 4 m ultiple root 2 0 -2 -4 0 1 2 x 3 4 Difficulties Bracketing methods won’t work Limited to methods that may diverge 10 8 f(x) 6 4 m ultiple root 2 0 -2 -4 0 1 2 x 3 4 denominator for Newton-Raphson and Secant Methods Write a “DO LOOP” to check is f(x) = 0 before continuing 10 8 6 f(x) f(x) = 0 at root f '(x) = 0 at root Hence, zero in the 4 m ultiple root 2 0 -2 -4 0 1 2 x 3 4 Multiple Roots f ' xi f xi f ' ' xi 2 10 8 6 f(x) xi 1 xi f xi f ' xi 4 m ultiple root 2 0 -2 -4 0 1 2 x 3 4 Systems of Non-Linear Equations We will later consider systems of linear equations f(x) = a1x1 + a2x2+...... anxn - C = 0 where a1 , a2 .... an and C are constant Consider the following equations y = -x2 + x + 0.5 y + 5xy = x3 Solve for x and y Systems of Non-Linear Equations cont. Set the equations equal to zero y = -x2 + x + 0.5 y + 5xy = x3 u(x,y) = -x2 + x + 0.5 - y = 0 v(x,y) = y + 5xy - x3 = 0 The solution would be the values of x and y that would make the functions u and v equal to zero Recall the Taylor Series f ' ' xi 2 f ' ' ' xi 3 f xi 1 f xi f ' xi h h h 2! 3! n f xi n ...... h Rn n! where h step size xi 1 xi Write a first order Taylor series with respect to u and v ui ui xi 1 xi yi 1 yi ui 1 ui x y vi vi xi 1 xi yi 1 yi vi 1 vi x y The root estimate corresponds to the point where ui+1 = vi+1 = 0 Therefore vi ui ui vi y y xi 1 xi ui vi ui vi x y y x ui vi vi ui x x yi 1 yi ui vi ui vi x y y x THE DENOMINATOR OF EACH OF THESE EQUATIONS IS FORMALLY REFERRED TO AS THE DETERMINANT OF THE JACOBIAN This is a 2 equation version of Newton-Raphson Example Determine the roots of the following nonlinear simultaneous equations x2+xy=10 y + 3xy2 = 57 Use and initial estimate of x=1.5, y=3.5 25 20 f(x) 15 10 5 STRATEGY 0 0 1 2 3 x 4 5 Strategy Rewrite equations to get u(x,y) = 0 from equation 1 v(x,y) = 0 from equation 2 Determine the equations for the partial of u and v with respect to x and y Start a table! i xi yi u (x,y) v(x,y) du/dx du/dy dv/dx dv/dy J Specific Study Objectives Understand the graphical interpretation of a root Know the graphical interpretation of the false-position method (regula falsi method) and why it is usually superior to the bisection method Understand the difference between bracketing and open methods for root location Specific Study Objectives Understand the concepts of convergence and divergence. Know why bracketing methods always converge, whereas open methods may sometimes diverge Know the fundamental difference between the false position and secant methods and how it relates to convergence Specific Study Objectives Understand the problems posed by multiple roots and the modification available to mitigate them Use the techniques presented to find the root of an equation Solve two nonlinear simultaneous equations Applied Problem The concentration of pollutant bacteria C in a lake decreases according to: C 80e2 t 20e0.1t Determine the time required for the bacteria to be reduced to 10 using Newton-Raphson method. Applied Problem You buy a $20 K piece of equipment for nothing down and $5K per year for 5 years. What interest rate are you paying? The formula relating present worth (P), annual payments (A), number of years (n) and the interest rate (i) is: i1 i A P 1 i n 1 n Use the bisection method