Week2 - Solving Linear System_3vars

numpy를 이용해 문제를 푸는 연습을 함

Linear equation을 Matrix로 표현하기 §

$$\{\begin{array}{l}4x_1-3x_2+x_3=-10,\\ 2x_1+x_2+3x_3=0,\\ -x_1+2x_2-5x_3=17,\end{array}\text{\hspace{1em}(1)}$$

A = np.array([
        [4, -3, 1],
        [2, 1, 3],
        [-1, 2, -5]
    ], dtype=np.dtype(float))
 
b = np.array([-10, 0, 17], dtype=np.dtype(float))
 
print("Matrix A:")
print(A)
print("\nArray b:")
print(b)

Matrix A:
[[ 4. -3.  1.]
 [ 2.  1.  3.]
 [-1.  2. -5.]]
 
Array b:
[-10.   0.  17.]

print(f"Shape of A: {np.shape(A)}")
print(f"Shape of b: {np.shape(b)}")

Shape of A: (3, 3)
Shape of b: (3,)

이 문제를 쉽게 푸는 방법은 아래와 같음

x = np.linalg.solve(A, b)
 
print(f"Solution: {x}")

Solution: [ 1.  4. -2.]

Determinant 계산 §

d = np.linalg.det(A)
 
print(f"Determinant of matrix A: {d:.2f}")

Determinant of matrix A: -60.00

Row reduction을 이용해 풀기 §

Row reduction 준비하기 §

hstack을 이용해 3*4 메트릭스 형태를 만듬

A_system = np.hstack((A, b.reshape((3, 1))))
 
print(A_system)

[[  4.  -3.   1. -10.]
 [  2.   1.   3.   0.]
 [ -1.   2.  -5.  17.]]

Elementary Operation §

# multiply row_num_1 by row_num_1_multiple and add it to the row_num_2, 
# exchanging row_num_2 of the matrix M in the result
def AddRows(M, row_num_1, row_num_2, row_num_1_multiple):
    M_new = M.copy()
    M_new[row_num_2] = row_num_1_multiple * M_new[row_num_1] + M_new[row_num_2]
    return M_new
 
print("Original matrix:")
print(A_system)
print("\nMatrix after exchange of the third row with the sum of itself and second row multiplied by 1/2:")
print(AddRows(A_system,1,2,1/2))

Original matrix:
[[  4.  -3.   1. -10.]
 [  2.   1.   3.   0.]
 [ -1.   2.  -5.  17.]]
 
Matrix after exchange of the third row with the sum of itself and second row multiplied by 1/2:
[[  4.   -3.    1.  -10. ]
 [  2.    1.    3.    0. ]
 [  0.    2.5  -3.5  17. ]]

# exchange row_num_1 and row_num_2 of the matrix M
def SwapRows(M, row_num_1, row_num_2):
    M_new = M.copy()
    M_new[[row_num_1, row_num_2]] = M_new[[row_num_2, row_num_1]]
    return M_new
 
print("Original matrix:")
print(A_system)
print("\nMatrix after exchange its first and third rows:")
print(SwapRows(A_system,0,2))

Original matrix:
[[  4.  -3.   1. -10.]
 [  2.   1.   3.   0.]
 [ -1.   2.  -5.  17.]]
 
Matrix after exchange its first and third rows:
[[ -1.   2.  -5.  17.]
 [  2.   1.   3.   0.]
 [  4.  -3.   1. -10.]]