Question

The Parks and Rec department has an opening for a Solver of Linear Equations. Since Leslie is great at solving systems of linear equations she zealously offers to give all candidates a quick demonstration. She finds the LU factorization of a 2000 x 2000 matrix M in 4096 seconds. She then takes another 256 seconds to apply both triangular substitutions (backward and forward) to solve the system of equations Mx = y for a given vector y. = The candidates are supposed to estimate (without actually running any code) the time it takes to factorize and efficiently solve a similar system of linear equations Ax = b, where A is a square matrix with dimension 500, given 30 different right-hand side b vectors. Provide your time estimate t.

Answer 1

Matrix are used to represent data in rows and columns

The estimated time to efficiently solve the square matrix A is 281 seconds

How to estimate the time to solve matrix A

The given parameters are:

$Matrix\ size = 2000\ by\ 2000$

$Time = 4096\ seconds$

$Substitution\ time = 25\ seconds$

The time to substitute the variables is fixed, however, the time to solve the matrix depends on the size of the matrix

The similar matrix is 1/16 of the original matrix of 2000 by 2000.

So, the time to factorize the matrix is:

$Time = \frac{1}{16} * 4096\ seconds$

$Time = 256\ seconds$

The estimate of the time to solve the new matrix is then calculated as follows:

$Total = 256\ seconds + 25\ seconds$

$Total = 281\ seconds$

Hence, the estimated time is 281 seconds

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