We consider the quadratic formula and how it relates to the limitations of floating point numbers....
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We consider the quadratic formula and how it relates to the limitations of floating point numbers. Often we find the roots of the equation az²+bx+c=0 (where we assume a 0). We might divide through by a to obtain the equation r²+ Br+C =0 (1) where B=b/a and C= c/a. We know that the roots of this equation, r₁ and r2, may be found by the formulas R₁ = R₂ = -B+√B²-4C 2 B-√B2-4C 2 (2) (3) For the sake of simplicity in this problem let us assume that B and C are positive numbers. We investigate what happens when B is much larger than C. (a) Suppose in equation (1) we take B = 10% and C=1-B-2. Find the two roots, r₁ and r2, of equation (1) with this choice of B and C by factoring it into the form (z-ri)(z-72). In this way we know the roots of this equation exactly. (b) What are the corresponding roots, call them R₁ and R₂ respectively, obtained by using formulas (2) and (3) in Matlab. (e) How well do R₁ and R₂ approximate ri and r2 using the relative error measure? To how many significant digits do R₁ and R₂ approximate ri and r2 respectively? (d) Consider the approximation R₁. Explain what causes this to be such a poor mumerical approximation to the actual root r₁. (e) What can be done to remedy this, i.e., what different formula or numerical technique might we use to fix this issue? Use this to find a better numerical approximation to r₁. What is this new approximation? To how many significant digits does it approximate ry? (f) Find the roots of equation (1) with B = 10% and C=2 to 10 significant digits. We consider the quadratic formula and how it relates to the limitations of floating point numbers. Often we find the roots of the equation az²+bx+c=0 (where we assume a 0). We might divide through by a to obtain the equation r²+ Br+C =0 (1) where B=b/a and C= c/a. We know that the roots of this equation, r₁ and r2, may be found by the formulas R₁ = R₂ = -B+√B²-4C 2 B-√B2-4C 2 (2) (3) For the sake of simplicity in this problem let us assume that B and C are positive numbers. We investigate what happens when B is much larger than C. (a) Suppose in equation (1) we take B = 10% and C=1-B-2. Find the two roots, r₁ and r2, of equation (1) with this choice of B and C by factoring it into the form (z-ri)(z-72). In this way we know the roots of this equation exactly. (b) What are the corresponding roots, call them R₁ and R₂ respectively, obtained by using formulas (2) and (3) in Matlab. (e) How well do R₁ and R₂ approximate ri and r2 using the relative error measure? To how many significant digits do R₁ and R₂ approximate ri and r2 respectively? (d) Consider the approximation R₁. Explain what causes this to be such a poor mumerical approximation to the actual root r₁. (e) What can be done to remedy this, i.e., what different formula or numerical technique might we use to fix this issue? Use this to find a better numerical approximation to r₁. What is this new approximation? To how many significant digits does it approximate ry? (f) Find the roots of equation (1) with B = 10% and C=2 to 10 significant digits.
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a If B 108 and C 1 B2 then equation 1 is x2 108x 1 1016 0 Factoring this we get x 108x 108 0 Therefo... View the full answer
Related Book For
Fundamentals Of Taxation 2015
ISBN: 9781259293092
8th Edition
Authors: Ana Cruz, Michael Deschamps, Frederick Niswander, Debra Prendergast, Dan Schisler, Jinhee Trone
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