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Please use MATLAB and explain your work for all 3 functions. Please do not use built-in functions, write out loops and simple operations (like addition, subtraction, multiplication, division)

Please use MATLAB and explain your work for all 3 functions. Please do not use built-in functions, write out loops and simple operations (like addition, subtraction, multiplication, division)

C++ for Engineers and Scientists
C++ for Engineers and Scientists
4th Edition
ISBN:9781133187844
Author:Bronson, Gary J.
Publisher:Bronson, Gary J.
Chapter6: Modularity Using Functions
Section6.4: A Case Study: Rectangular To Polar Coordinate Conversion
Problem 6E
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Please use MATLAB and explain your work for all 3 functions. Please do not use built-in functions, write out loops and simple operations (like addition, subtraction, multiplication, division). Thank you!

### Function #3 **Inputs:** - Matrix L - Matrix U - Vector b **Output:** - Vector x **Task:** Using forward and backward substitution, find the x vector for the equation \( LUx = b \) ---- Function #1 should produce the same results (within a reasonable amount of rounding errors) of the process involving Function #2 & #3. Once you believe your functions are working, test the time difference when provided problems such as: \[ Ax = b1, Ax = b2, Ax = b3, Ax = b4 \] where \( b1, b2, b3, b4 \) are different b vectors to solve the equation with.
Transcribed Image Text:### Function #3 **Inputs:** - Matrix L - Matrix U - Vector b **Output:** - Vector x **Task:** Using forward and backward substitution, find the x vector for the equation \( LUx = b \) ---- Function #1 should produce the same results (within a reasonable amount of rounding errors) of the process involving Function #2 & #3. Once you believe your functions are working, test the time difference when provided problems such as: \[ Ax = b1, Ax = b2, Ax = b3, Ax = b4 \] where \( b1, b2, b3, b4 \) are different b vectors to solve the equation with.
### Function #1: **Inputs:** Matrix A, vector b **Output:** vector x **Task:** Solve for \(x\) in the equation \(Ax = b\), using Gaussian elimination and permutations. Use row-reduction techniques to get the matrix into an identity matrix form (while performing the same operations on the b vector), so that the end version of the b vector becomes your x vector. --- ### Function #2: **Inputs:** Matrix A **Output:** Matrix L, Matrix U **Task:** Find the LU Factorization of matrix A --- In Function #1, the goal is to solve the linear equation \(Ax = b\). This involves transforming matrix \(A\) into an identity matrix through Gaussian elimination, which simplifies the equations so that vector \(b\) can be transformed into vector \(x\). In Function #2, the task involves breaking down matrix \(A\) into its LU Decomposition, where \(L\) and \(U\) are lower and upper triangular matrices, respectively. This is a fundamental method in numerical linear algebra for solving matrices more efficiently. Understanding these processes is essential for students studying linear algebra, as they form the basis for solving complex systems of equations computationally.
Transcribed Image Text:### Function #1: **Inputs:** Matrix A, vector b **Output:** vector x **Task:** Solve for \(x\) in the equation \(Ax = b\), using Gaussian elimination and permutations. Use row-reduction techniques to get the matrix into an identity matrix form (while performing the same operations on the b vector), so that the end version of the b vector becomes your x vector. --- ### Function #2: **Inputs:** Matrix A **Output:** Matrix L, Matrix U **Task:** Find the LU Factorization of matrix A --- In Function #1, the goal is to solve the linear equation \(Ax = b\). This involves transforming matrix \(A\) into an identity matrix through Gaussian elimination, which simplifies the equations so that vector \(b\) can be transformed into vector \(x\). In Function #2, the task involves breaking down matrix \(A\) into its LU Decomposition, where \(L\) and \(U\) are lower and upper triangular matrices, respectively. This is a fundamental method in numerical linear algebra for solving matrices more efficiently. Understanding these processes is essential for students studying linear algebra, as they form the basis for solving complex systems of equations computationally.
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