Question

KVL equations of circuit can be written as

15I₁ −3I₂ −8I₃ −I₄ =25
−3I₁ +25I₂ −2I₃ −6I₄ =2
−8V₁ −2V₂ +11I₃ −4I₄ =15
−V₁ −6V₂ −4I₃174I₄ =−5
​
Write is in form of A⋅X=b and solve it with LU decomposition method
a. Decompose the coefficient matrix as L and U

Answer 1

The coefficient matrix as L and U:

L = [[1.0, 0.0, 0.0, 0.0],

[-0.2, 1.0, 0.0, 0.0],

[-0.53333333, 0.04, 1.0, 0.0],

[-0.06666667, -0.4, 0.23529412, 1.0]]

U = [[15.0, -3.0, -8.0, -1.0],

[0.0, 25

Step-by-step explanation:

Part a: Decomposing the coefficient matrix as L and U

Here is the solution for part a without using Python code:

LU decomposition steps:

1. Initialize L and U matrices:

• Create two matrices, L and U, of the same size as the coefficient matrix A.
• Initialize L to the identity matrix and U to a zero matrix.

2. Gaussian elimination:

For each pivot column k (1 to n-1):

- For each row i below k (k+1 to n):

- Calculate the multiplier `m_ik = A[i, k] / A[k, k]`.

- Update L: `L[i, k] = m_ik`.

- Update A[i, j] for all j (k+1 to n): `A[i, j] -= m_ik * A[k, j]`.

3. Copy remaining diagonal elements to U:

For each i (k to n):

• Set U[i, i] = A[i, i].

Applying the steps to the given matrix:

1. Initialize L and U:

L = [[1, 0, 0, 0],

[0, 1, 0, 0],

[0, 0, 1, 0],

[0, 0, 0, 1]]

U = [[0, 0, 0, 0],

[0, 0, 0, 0],

[0, 0, 0, 0],

[0, 0, 0, 0]]

2. Gaussian elimination:

Pivot column 1:

- Update row 2:

`L[2, 1] = -3/15 = -0.2`

`A[2, 1:] = [-3, 25, -2, -6] - 0.2 * [15, -3, -8, -1]`

= [0, 25, -1.86666667, -5.73333333]

- Update row 3:

`L[3, 1] = -8/15 = -0.53333333`

`A[3, 1:] = [-8, -2, 11, -4] - 0.53333333 * [15, -3, -8, -1]`

= [0, 0, 10.40625, -3.52941176]

- Update U:

`U[1, 1] = 15`

Pivot column 2:

- Update row 3:

`L[3, 2] = 0.04`

`A[3, 3:] = [10.40625, -3.52941176] - 0.04 * [25, -6]`

= [10.40625, -3.64705882]

- Update U:

`U[2, 2] = 25`

`U[2, 3] = -1.86666667`

`U[2, 4] = -5.73333333`

Pivot column 3:

- Update U:

`U[3, 3] = 10.40625`

`U[3, 4] = -3.52941176`

3. Final L and U matrices:

L = [[1.0, 0.0, 0.0, 0.0],

[-0.2, 1.0, 0.0, 0.0],

[-0.53333333, 0.04, 1.0, 0.0],

[-0.06666667, -0.4, 0.23529412, 1.0]]

U = [[15.0, -3.0, -8.0, -1.0],

[0.0, 25