## What is the most computationally efficient factorization of a matrix A?

I have an exam in 24 hours for my Linear Algebra and Geometry course and we have a section with MATLAB related questions. In a simulation test, I was asked to "solve the linear system Ax=b by the solution of two triangular systems, using the most computationally efficient factorization of a matrix A". What does this mean? What is "the solution of two triangular systems"? How would I choose the most computationally efficient factorization of a matrix?

The question I am talking about is the following, word for word:

Let Ax = b, a linear system of order 18, where A is a symmetric, tridiagonal matrix with all elements equal to 6 on the main diagonal and to 3 on the superior and inferior codiagonals. The elements of b are linearly spaced numbers in [8,11]. Compute the eigenvalues of A and, based on their properties, solve the linear system Ax = b by the solution of two triangular systems, using the most computationally efficient factorization of A. The 1-norm of the vector obtained by summing the vector that is the solution of the inferior triangular system and the one that is the solution of the superior triangular system is, approximately:

A) 8.4419e+01

B) 3.4638e+00

C) 7.5558e-01

D) 9.5155e-01

E) 6.5559e+01

Now, we went over all this with a friend and even asked chatgpt to give us an idea. In our curriculum, the main factorizations we learned about are the PA=LU, the Choleski, SVD and the QR factorization. We thought maybe after seeing all the eigenvalues are positive, and thus the matrix positive definite, the choleski should be used. But we aren’t really sure. Why would the others be less efficient? With the code chatgpt gave us, tweaked a little here and there, we got the answer right using the choleski. But we really aren’t sure we understand the wording of the question and the systems at play. Here is the code in question:

Thank you in advance for an answer to such a long question.

clear, clc

n = 18;

A = eye(n)*6 + diag(ones(n-1,1), 1)*3 + diag(ones(n-1,1), -1)*3;

b = linspace(8,11,18);

eigens = eig(A)

R = chol(A);

disp("R’*R*x = b so y = R*x = b")

Y = R’b’;

x = RY;

finalVector = x + Y;

norm(finalVector, 1)I have an exam in 24 hours for my Linear Algebra and Geometry course and we have a section with MATLAB related questions. In a simulation test, I was asked to "solve the linear system Ax=b by the solution of two triangular systems, using the most computationally efficient factorization of a matrix A". What does this mean? What is "the solution of two triangular systems"? How would I choose the most computationally efficient factorization of a matrix?

The question I am talking about is the following, word for word:

Let Ax = b, a linear system of order 18, where A is a symmetric, tridiagonal matrix with all elements equal to 6 on the main diagonal and to 3 on the superior and inferior codiagonals. The elements of b are linearly spaced numbers in [8,11]. Compute the eigenvalues of A and, based on their properties, solve the linear system Ax = b by the solution of two triangular systems, using the most computationally efficient factorization of A. The 1-norm of the vector obtained by summing the vector that is the solution of the inferior triangular system and the one that is the solution of the superior triangular system is, approximately:

A) 8.4419e+01

B) 3.4638e+00

C) 7.5558e-01

D) 9.5155e-01

E) 6.5559e+01

Now, we went over all this with a friend and even asked chatgpt to give us an idea. In our curriculum, the main factorizations we learned about are the PA=LU, the Choleski, SVD and the QR factorization. We thought maybe after seeing all the eigenvalues are positive, and thus the matrix positive definite, the choleski should be used. But we aren’t really sure. Why would the others be less efficient? With the code chatgpt gave us, tweaked a little here and there, we got the answer right using the choleski. But we really aren’t sure we understand the wording of the question and the systems at play. Here is the code in question:

Thank you in advance for an answer to such a long question.

clear, clc

n = 18;

A = eye(n)*6 + diag(ones(n-1,1), 1)*3 + diag(ones(n-1,1), -1)*3;

b = linspace(8,11,18);

eigens = eig(A)

R = chol(A);

disp("R’*R*x = b so y = R*x = b")

Y = R’b’;

x = RY;

finalVector = x + Y;

norm(finalVector, 1) I have an exam in 24 hours for my Linear Algebra and Geometry course and we have a section with MATLAB related questions. In a simulation test, I was asked to "solve the linear system Ax=b by the solution of two triangular systems, using the most computationally efficient factorization of a matrix A". What does this mean? What is "the solution of two triangular systems"? How would I choose the most computationally efficient factorization of a matrix?

The question I am talking about is the following, word for word:

Let Ax = b, a linear system of order 18, where A is a symmetric, tridiagonal matrix with all elements equal to 6 on the main diagonal and to 3 on the superior and inferior codiagonals. The elements of b are linearly spaced numbers in [8,11]. Compute the eigenvalues of A and, based on their properties, solve the linear system Ax = b by the solution of two triangular systems, using the most computationally efficient factorization of A. The 1-norm of the vector obtained by summing the vector that is the solution of the inferior triangular system and the one that is the solution of the superior triangular system is, approximately:

A) 8.4419e+01

B) 3.4638e+00

C) 7.5558e-01

D) 9.5155e-01

E) 6.5559e+01

Now, we went over all this with a friend and even asked chatgpt to give us an idea. In our curriculum, the main factorizations we learned about are the PA=LU, the Choleski, SVD and the QR factorization. We thought maybe after seeing all the eigenvalues are positive, and thus the matrix positive definite, the choleski should be used. But we aren’t really sure. Why would the others be less efficient? With the code chatgpt gave us, tweaked a little here and there, we got the answer right using the choleski. But we really aren’t sure we understand the wording of the question and the systems at play. Here is the code in question:

Thank you in advance for an answer to such a long question.

clear, clc

n = 18;

A = eye(n)*6 + diag(ones(n-1,1), 1)*3 + diag(ones(n-1,1), -1)*3;

b = linspace(8,11,18);

eigens = eig(A)

R = chol(A);

disp("R’*R*x = b so y = R*x = b")

Y = R’b’;

x = RY;

finalVector = x + Y;

norm(finalVector, 1) efficiency, factorization MATLAB Answers — New Questions