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Consider a system Ux=b, where U is upper triangular. a) Modify the solution we developed in class for lower triangular systems to upper triangular systems (in the upper triangular case, this process is called backward substitution). (In particular, find the formula to compute x, given this time Xi+1,...,xn.) b) Using the model code for lowtriangsolve.m, write a MATLAB code that solves unnor triangulor gustoms

Consider a system Ux=b, where U is upper triangular. a) Modify the solution we developed in class for lower triangular systems to upper triangular systems (in the upper triangular case, this process is called backward substitution). (In particular, find the formula to compute x, given this time Xi+1,...,xn.) b) Using the model code for lowtriangsolve.m, write a MATLAB code that solves unnor triangulor gustoms

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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Suppose the invertible matrix A has an LU factorization. Prove that this factorization is unique by considering the possibility that LU = ŽŨ, for some L, Î lower triangular matrices with all 1s on the diagonal, and U,Ũ upper triangular matrices, such that L ‡ Ĩ or U ‡ Ũ. (Hint: you will need Q3, and Problem Q4 from HW 1.)
Transcribed Image Text:Suppose the invertible matrix A has an LU factorization. Prove that this factorization is unique by considering the possibility that LU = ŽŨ, for some L, Î lower triangular matrices with all 1s on the diagonal, and U,Ũ upper triangular matrices, such that L ‡ Ĩ or U ‡ Ũ. (Hint: you will need Q3, and Problem Q4 from HW 1.)
Q3. (This is Exercise 1.3.15 from the textbook) Consider a system Ux=b, where U is upper triangular. a) Modify the solution we developed in class for lower triangular systems to upper triangular systems (in the upper triangular case, this process is called backward substitution). (In particular, find the formula to compute xį given this time Xi+1,. xn.) b) Using the model code for lowtriangsolve.m, write a MATLAB code that solves upper triangular systems. Q4. Let U be an invertible upper triangular n × n matrix. ... 9 a) Let b a vector of length n with b; ‡ 0, and bi+1 = bi+2 1 ≤ i ≤n. Let x be the unique solution to the system Ux = b. By using backward substitution, show that x has the same pattern of zeros as b, that is, x; 0, and Xi+1 = Xi+2 = = ... = Χη = 0. b) Prove that U-¹ is also upper triangular. Hint: Note that the j-th column x; of U-¹ satisfies 9 = U xj = ¤j, j = 1, where n is the size of the matrix and e; is the j-th unit vector, with a 1 in position j and Os everywhere else. Then, use a). 1 n = = bn = 0, for some
Transcribed Image Text:Q3. (This is Exercise 1.3.15 from the textbook) Consider a system Ux=b, where U is upper triangular. a) Modify the solution we developed in class for lower triangular systems to upper triangular systems (in the upper triangular case, this process is called backward substitution). (In particular, find the formula to compute xį given this time Xi+1,. xn.) b) Using the model code for lowtriangsolve.m, write a MATLAB code that solves upper triangular systems. Q4. Let U be an invertible upper triangular n × n matrix. ... 9 a) Let b a vector of length n with b; ‡ 0, and bi+1 = bi+2 1 ≤ i ≤n. Let x be the unique solution to the system Ux = b. By using backward substitution, show that x has the same pattern of zeros as b, that is, x; 0, and Xi+1 = Xi+2 = = ... = Χη = 0. b) Prove that U-¹ is also upper triangular. Hint: Note that the j-th column x; of U-¹ satisfies 9 = U xj = ¤j, j = 1, where n is the size of the matrix and e; is the j-th unit vector, with a 1 in position j and Os everywhere else. Then, use a). 1 n = = bn = 0, for some
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