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Consider the problem of finding a quadratic polynomial p(x) which best fits the data (-1,0),(0,1),(1,2),(2,4). b the four equations p(-1) = 0,p(0) = (a) Write in matrix form Ac = 1,p(1) = 2, p(2) = 4. (b) Is there a solution c to the matrix equation Ac = b in part (a)? (c) Write down the normal equation of Ac = b. (d) Without calculating the solutions to this normal equation, explain why it must have at least one solution.

Consider the problem of finding a quadratic polynomial p(x) which best fits the data (-1,0),(0,1),(1,2),(2,4). b the four equations p(-1) = 0,p(0) = (a) Write in matrix form Ac = 1,p(1) = 2, p(2) = 4. (b) Is there a solution c to the matrix equation Ac = b in part (a)? (c) Write down the normal equation of Ac = b. (d) Without calculating the solutions to this normal equation, explain why it must have at least one solution.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** Consider the problem of finding a quadratic polynomial \( p(x) \) which best fits the data points \((-1,0), (0,1), (1,2), (2,4)\). **Questions:** (a) Write in matrix form \( Ac = b \) the four equations \( p(-1) = 0, p(0) = 1, p(1) = 2, p(2) = 4 \). (b) Is there a solution \( \mathbf{c} \) to the matrix equation \( Ac = b \) in part (a)? (c) Write down the normal equation of \( Ac = b \). (d) Without calculating the solutions to this normal equation, explain why it must have at least one solution.
Transcribed Image Text:**Problem Statement:** Consider the problem of finding a quadratic polynomial \( p(x) \) which best fits the data points \((-1,0), (0,1), (1,2), (2,4)\). **Questions:** (a) Write in matrix form \( Ac = b \) the four equations \( p(-1) = 0, p(0) = 1, p(1) = 2, p(2) = 4 \). (b) Is there a solution \( \mathbf{c} \) to the matrix equation \( Ac = b \) in part (a)? (c) Write down the normal equation of \( Ac = b \). (d) Without calculating the solutions to this normal equation, explain why it must have at least one solution.
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