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1. Consider a set of four data points: f(0) = 3, f(4) = -2, f(-1) = 2, and f(1) = 1. In the following, use these data points to find the best fit polynomial of degree 2 by using the QR-decomposition method: (a) Identify the matrix A and b. Now, write down the linearly independent column vectors u₁, u2 and u3 from the matrix A.

1. Consider a set of four data points: f(0) = 3, f(4) = -2, f(-1) = 2, and f(1) = 1. In the following, use these data points to find the best fit polynomial of degree 2 by using the QR-decomposition method: (a) Identify the matrix A and b. Now, write down the linearly independent column vectors u₁, u2 and u3 from the matrix A.

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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1. Consider a set of four data points: f(0) = 3, f(4) = -2, f(-1) = 2, and f(1) = 1. In the following, use these data points to find the best fit polynomial of degree 2 by using the QR-decomposition method: (a) Identify the matrix A and b. Now, write down the linearly independent column vectors u₁, ₂ and u3 from the matrix A. (b) Using the Gram-Schmidt process construct the orthonormal column matrices (or vectors) 91, 92 and q3 from the linearly independent column vectors obtained in the previous part, and then write down the Q matrix. 예de Now calculate the matrix elements of R, and write down the matrix R. Compute Rx and QTb, where x = (ao a₁ a2) which are the coefficients of the polynomial p2(x). Using the above result, find the values of ao, a₁ and a2, and write the polynomial p2(x).
Transcribed Image Text:1. Consider a set of four data points: f(0) = 3, f(4) = -2, f(-1) = 2, and f(1) = 1. In the following, use these data points to find the best fit polynomial of degree 2 by using the QR-decomposition method: (a) Identify the matrix A and b. Now, write down the linearly independent column vectors u₁, ₂ and u3 from the matrix A. (b) Using the Gram-Schmidt process construct the orthonormal column matrices (or vectors) 91, 92 and q3 from the linearly independent column vectors obtained in the previous part, and then write down the Q matrix. 예de Now calculate the matrix elements of R, and write down the matrix R. Compute Rx and QTb, where x = (ao a₁ a2) which are the coefficients of the polynomial p2(x). Using the above result, find the values of ao, a₁ and a2, and write the polynomial p2(x).
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Follow-up Question

solve the question no d and e 

1. Consider a set of four data points: f(0) = 3, f(4) = -2, f(-1) = 2, and f(1) = 1. In the following, use these data points to find the best fit polynomial of degree 2 by using the QR-decomposition method: (a) Identify the matrix A and b. Now, write down the linearly independent column vectors u₁, ₂ and u3 from the matrix A. (b) Using the Gram-Schmidt process construct the orthonormal column matrices (or vectors) 91, 92 and q3 from the linearly independent column vectors obtained in the previous part, and then write down the Q matrix. 예de Now calculate the matrix elements of R, and write down the matrix R. Compute Rx and QTb, where x = (ao a₁ a2) which are the coefficients of the polynomial p2(x). Using the above result, find the values of ao, a₁ and a2, and write the polynomial p2(x).
Transcribed Image Text:1. Consider a set of four data points: f(0) = 3, f(4) = -2, f(-1) = 2, and f(1) = 1. In the following, use these data points to find the best fit polynomial of degree 2 by using the QR-decomposition method: (a) Identify the matrix A and b. Now, write down the linearly independent column vectors u₁, ₂ and u3 from the matrix A. (b) Using the Gram-Schmidt process construct the orthonormal column matrices (or vectors) 91, 92 and q3 from the linearly independent column vectors obtained in the previous part, and then write down the Q matrix. 예de Now calculate the matrix elements of R, and write down the matrix R. Compute Rx and QTb, where x = (ao a₁ a2) which are the coefficients of the polynomial p2(x). Using the above result, find the values of ao, a₁ and a2, and write the polynomial p2(x).
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Follow-up Question

solve c and d 

1. Consider a set of four data points: f(0) = 3, f(4) = -2, f(-1) = 2, and f(1) = 1. In the following, use these data points to find the best fit polynomial of degree 2 by using the QR-decomposition method: (a) Identify the matrix A and b. Now, write down the linearly independent column vectors u₁, ₂ and u3 from the matrix A. (b) Using the Gram-Schmidt process construct the orthonormal column matrices (or vectors) 91, 92 and q3 from the linearly independent column vectors obtained in the previous part, and then write down the Q matrix. 예de Now calculate the matrix elements of R, and write down the matrix R. Compute Rx and QTb, where x = (ao a₁ a2) which are the coefficients of the polynomial p2(x). Using the above result, find the values of ao, a₁ and a2, and write the polynomial p2(x).
Transcribed Image Text:1. Consider a set of four data points: f(0) = 3, f(4) = -2, f(-1) = 2, and f(1) = 1. In the following, use these data points to find the best fit polynomial of degree 2 by using the QR-decomposition method: (a) Identify the matrix A and b. Now, write down the linearly independent column vectors u₁, ₂ and u3 from the matrix A. (b) Using the Gram-Schmidt process construct the orthonormal column matrices (or vectors) 91, 92 and q3 from the linearly independent column vectors obtained in the previous part, and then write down the Q matrix. 예de Now calculate the matrix elements of R, and write down the matrix R. Compute Rx and QTb, where x = (ao a₁ a2) which are the coefficients of the polynomial p2(x). Using the above result, find the values of ao, a₁ and a2, and write the polynomial p2(x).
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Follow-up Question

try to solve b c and d 

1. Consider a set of four data points: f(0) = 3, f(4) = -2, f(-1) = 2, and f(1) = 1. In the following, use these data points to find the best fit polynomial of degree 2 by using the QR-decomposition method: (a) Identify the matrix A and b. Now, write down the linearly independent column vectors u₁, ₂ and u3 from the matrix A. (b) Using the Gram-Schmidt process construct the orthonormal column matrices (or vectors) 91, 92 and q3 from the linearly independent column vectors obtained in the previous part, and then write down the Q matrix. 예de Now calculate the matrix elements of R, and write down the matrix R. Compute Rx and QTb, where x = (ao a₁ a2) which are the coefficients of the polynomial p2(x). Using the above result, find the values of ao, a₁ and a2, and write the polynomial p2(x).
Transcribed Image Text:1. Consider a set of four data points: f(0) = 3, f(4) = -2, f(-1) = 2, and f(1) = 1. In the following, use these data points to find the best fit polynomial of degree 2 by using the QR-decomposition method: (a) Identify the matrix A and b. Now, write down the linearly independent column vectors u₁, ₂ and u3 from the matrix A. (b) Using the Gram-Schmidt process construct the orthonormal column matrices (or vectors) 91, 92 and q3 from the linearly independent column vectors obtained in the previous part, and then write down the Q matrix. 예de Now calculate the matrix elements of R, and write down the matrix R. Compute Rx and QTb, where x = (ao a₁ a2) which are the coefficients of the polynomial p2(x). Using the above result, find the values of ao, a₁ and a2, and write the polynomial p2(x).
Solution
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Follow-up Question
1. Consider a set of four data points: f(0) = 3, f(4) = -2, f(-1) = 2, and f(1) = 1. In the following, use these data points to find the best fit polynomial of degree 2 by using the QR-decomposition method: (a) Identify the matrix A and b. Now, write down the linearly independent column vectors u₁, ₂ and u3 from the matrix A. (b) Using the Gram-Schmidt process construct the orthonormal column matrices (or vectors) 91, 92 and q3 from the linearly independent column vectors obtained in the previous part, and then write down the Q matrix. 예de Now calculate the matrix elements of R, and write down the matrix R. Compute Rx and QTb, where x = (ao a₁ a2) which are the coefficients of the polynomial p2(x). Using the above result, find the values of ao, a₁ and a2, and write the polynomial p2(x).
Transcribed Image Text:1. Consider a set of four data points: f(0) = 3, f(4) = -2, f(-1) = 2, and f(1) = 1. In the following, use these data points to find the best fit polynomial of degree 2 by using the QR-decomposition method: (a) Identify the matrix A and b. Now, write down the linearly independent column vectors u₁, ₂ and u3 from the matrix A. (b) Using the Gram-Schmidt process construct the orthonormal column matrices (or vectors) 91, 92 and q3 from the linearly independent column vectors obtained in the previous part, and then write down the Q matrix. 예de Now calculate the matrix elements of R, and write down the matrix R. Compute Rx and QTb, where x = (ao a₁ a2) which are the coefficients of the polynomial p2(x). Using the above result, find the values of ao, a₁ and a2, and write the polynomial p2(x).
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