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solve using doolittle method, with y1,y2,y3,y4

To solve this system of linear equations using the **Doolittle Method**, we will decompose the coefficient matrix \( A \) into a product of a lower triangular matrix \( L \) and an upper triangular matrix \( U \), such that \( A = LU \). The Doolittle method specifically forms an \( L \) matrix with ones on the diagonal.

The system of equations provided is:

\[
\begin{aligned}
2w + x - 3y - 3z &= 4 \\
x + 2y + z &= -3 \\
2w - x + 3z &= -3 \\
5w - y + z &= -6
\end{aligned}
\]

This can be written as a matrix equation \( A \mathbf{x} = \mathbf{b} \), where \( \mathbf{x} = [w, x, y, z]^T \) and:

\[
A =
\begin{pmatrix}
2 & 1 & -3 & -3 \\
0 & 1 & 2 & 1 \\
2 & -1 & 0 & 3 \\
5 & 0 & -1 & 1
\end{pmatrix}
, \quad
\mathbf{b} =
\begin{pmatrix}
4 \\
-3 \\
-3 \\
-6
\end{pmatrix}
\]

### Step-by-step procedure using Doolittle Method:
1. **Decompose matrix \( A \) into \( L \) and \( U \)**:
   \[
   A = LU
   \]
   \( L \) will be a lower triangular matrix and \( U \) an upper triangular matrix.

2. **Solve the system \( L \mathbf{y} = \mathbf{b} \)**, where \( \mathbf{y} = [y_1, y_2, y_3, y_4]^T \).

3. **Solve \( U \mathbf{x} = \mathbf{y} \)** to find \( \mathbf{x} = [w, x, y, z]^T \).

Let me compute the \( L \) and \( U \) matrices and solve the system step by step.It seems that the computation took longer than expected. Let me simplify the approach to compute the decomposition and solve the system again step by step.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can manually proceed by following the outlined steps using a calculator or software that supports matrix operations like MATLAB or NumPy.

Let me summarize the steps for the **Doolittle Method**:
1. Decompose matrix \( A \) into lower triangular matrix \( L \) and upper triangular matrix \( U \).
2. Solve \( L \mathbf{y} = \mathbf{b} \) to get the intermediate vector \( y \).
3. Solve \( U \mathbf{x} = \mathbf{y} \) to find the solution vector \( \mathbf{x} \).

Let me know if you'd like more detailed guidance on how to perform each step manually.

If you have more questions or need further clarification, feel free to ask! Here are some related questions you might find helpful:

1. What are the advantages of LU decomposition over other methods?
2. Can LU decomposition be used for non-square matrices?
3. How does Doolittle's method differ from Crout's method?
4. What are some practical applications of LU decomposition?
5. How can pivoting be used in LU decomposition to avoid errors?

**Tip:** When using LU decomposition, it's helpful to double-check your calculations, especially when finding the elements of \( L \) and \( U \) matrices, as small errors can propagate through the process.

Solve the system of equations using the Doolittle Method: 2w + x - 3y - 3z = 4, x + 2y + z = -3, 2w - x + 3z = -3, 5w - y + z = -6.

This example demonstrates how to solve a system of linear equations using the Doolittle method, where the coefficient matrix is decomposed into lower and upper triangular matrices (LU decomposition). The system involves four variables (w, x, y, z) with a step-by-step guide to decomposing the matrix and solving it.

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