1. (a) Consider the polynomial (x − 1)(x − 2) = x² -3x+2 which has exact roots x = 1 and x = 2. Suppose some numerical algorithm finds approximate roots equal to ✰ = 1.1 and = 1.9. For each root determine the forward error and the relative forward error. Then, for each approximate root, find a perturbed quadratic, that is as close to the original quadratic as possible, such that the approximate root to the original quadratic is the exact root to the perturbed quadratic. For each approximate root, what is the backward error and what is the relative backward error? (b) For each approximate root, draw a diagram like the ones we've seen in the course notes, showing the relationship between the original problem the perturbed problem, the exact root, and the approximate root. Make sure to include the value of the forward error and the backward error in the diagram.

Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
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1. (a) Consider the polynomial (x − 1)(x − 2) = x² -3x+2 which has exact roots x = 1 and
x = 2. Suppose some numerical algorithm finds approximate roots equal to ✰ = 1.1 and
= 1.9. For each root determine the forward error and the relative forward error. Then,
for each approximate root, find a perturbed quadratic, that is as close to the original
quadratic as possible, such that the approximate root to the original quadratic is the
exact root to the perturbed quadratic. For each approximate root, what is the backward
error and what is the relative backward error?
(b) For each approximate root, draw a diagram like the ones we've seen in the course
notes, showing the relationship between the original problem the perturbed problem, the
exact root, and the approximate root. Make sure to include the value of the forward error
and the backward error in the diagram.
Transcribed Image Text:1. (a) Consider the polynomial (x − 1)(x − 2) = x² -3x+2 which has exact roots x = 1 and x = 2. Suppose some numerical algorithm finds approximate roots equal to ✰ = 1.1 and = 1.9. For each root determine the forward error and the relative forward error. Then, for each approximate root, find a perturbed quadratic, that is as close to the original quadratic as possible, such that the approximate root to the original quadratic is the exact root to the perturbed quadratic. For each approximate root, what is the backward error and what is the relative backward error? (b) For each approximate root, draw a diagram like the ones we've seen in the course notes, showing the relationship between the original problem the perturbed problem, the exact root, and the approximate root. Make sure to include the value of the forward error and the backward error in the diagram.
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