EBK NUMERICAL METHODS FOR ENGINEERS
7th Edition
ISBN: 8220100254147
Author: Chapra
Publisher: MCG
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Chapter 18, Problem 8P
The following data come from a table that was measured with high precision. Use the best numerical method (for this type of problem to determine y at
| x | 0 | 1.8 | 5 | 6 | 8.2 | 9.2 | 12 |
| y | 26 | 16.415 | 5.375 | 3.5 | 2.015 | 2.54 | 8 |
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Your work should be submitted on Moodle, before February 7 at 5 pm.
1. True or false:
(a) if E is a subspace of V, then dim(E) + dim(E) = dim(V)
(b) Let {i, n} be a basis of the vector space V, where v₁,..., Un are all eigen-
vectors for both the matrix A and the matrix B. Then, any eigenvector of A is
an eigenvector of B.
Justify.
2. Apply Gram-Schmidt orthogonalization to the system of vectors {(1,2,-2), (1, −1, 4), (2, 1, 1)}.
3. Suppose P is the orthogonal projection onto a subspace E, and Q is the orthogonal
projection onto the orthogonal complement E.
(a) The combinations of projections P+Q and PQ correspond to well-known oper-
ators. What are they? Justify your answer.
(b) Show…
1. True or false:
(a) if E is a subspace of V, then dim(E) + dim(E+) = dim(V)
(b) Let {i, n} be a basis of the vector space V, where vi,..., are all eigen-
vectors for both the matrix A and the matrix B. Then, any eigenvector of A is
an eigenvector of B.
Justify.
2. Apply Gram-Schmidt orthogonalization to the system of vectors {(1, 2, -2), (1, −1, 4), (2, 1, 1)}.
3. Suppose P is the orthogonal projection onto a subspace E, and Q is the orthogonal
projection onto the orthogonal complement E.
(a) The combinations of projections P+Q and PQ correspond to well-known oper-
ators. What are they? Justify your answer.
(b) Show that P - Q is its own inverse.
4. Show that the Frobenius product on n x n-matrices,
(A, B) =
= Tr(B*A),
is an inner product, where B* denotes the Hermitian adjoint of B.
5. Show that if A and B are two n x n-matrices for which {1,..., n} is a basis of eigen-
vectors (for both A and B), then AB = BA.
Remark: It is also true that if AB = BA, then there exists a common…
Chapter 18 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
Ch. 18 - 18.1 Estimate the common logarithm of 10 using...Ch. 18 - 18.2 Fit a second-order Newton’s interpolating...Ch. 18 - 18.3 Fit a third-order Newton’s interpolating...Ch. 18 - Repeat Probs. 18.1 through 18.3 using the Lagrange...Ch. 18 - 18.5 Given these...Ch. 18 - 18.6 Given these data
x 1 2 3 5 7 8
...Ch. 18 - Repeat Prob. 18.6 using Lagrange polynomials of...Ch. 18 - 18.8 The following data come from a table that was...Ch. 18 - 18.9 Use Newton’s interpolating polynomial to...Ch. 18 - Use Newtons interpolating polynomial to determine...
Ch. 18 - 18.11 Employ inverse interpolation using a cubic...Ch. 18 - 18.12 Employ inverse interpolation to determine...Ch. 18 - 18.13 Develop quadratic splines for the first five...Ch. 18 - 18.14 Develop cubic splines for the data in Prob....Ch. 18 - Determine the coefficients of the parabola that...Ch. 18 - Determine the coefficients of the cubic equation...Ch. 18 - 18.17 Develop, debug, and test a program in either...Ch. 18 - 18.18 Test the program you developed in Prob....Ch. 18 - 18.19 Use the program you developed in Prob. 18.17...Ch. 18 - Use the program you developed in Prob. 18.17 to...Ch. 18 - Develop, debug, and test a program in either a...Ch. 18 - 18.22 A useful application of Lagrange...Ch. 18 - Develop, debug, and test a program in either a...Ch. 18 - Use the software developed in Prob. 18.23 to fit...Ch. 18 - Use the portion of the given steam table for...Ch. 18 - The following is the built-in humps function that...Ch. 18 - 18.28 The following data define the sea-level...Ch. 18 - 18.29 Generate eight equally-spaced points from...Ch. 18 - 18.30 Temperatures are measured at various points...
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