[2022 Fall] Assignment6-6¶

Course: AP3021

Q6-6¶

1. without relaxation.¶

In [1]:

import numpy as np

import numpy as np

In [2]:

a = np.array([[ 2.0, -6.0, -1.0],
              [-3.0, -1.0,  7.0],
              [-8.0,  1.0, -2.0]])
b = np.array([[-38.0],
              [-34.0],
              [-20.0]])

def pivot(a, b):  

    p = np.array((a), float)
    q = np.array((b), float)
    n = len(q)

    for i in range(0, n - 1):

        if abs(p[i, 1]) == 0:
            for k in range(i + 1, n):
                if abs((p[k, i])) > abs(p[i, i]):
                    p[[i, k]] = p[[k, i]]
                    q[[i, k]] = q[[k, i]]

                    break

        for j in range(i + 1, n):
            f = p[j][i] / p[i][i]
            p[j, :] = p[j, :] - f * p[i, :]
            q[j] = q[j] -f * q[i]

    return p,q


a, b = pivot(a, b)
x1 = 1
x2 = 1
x3 = 1
count = 0
es = 0.05

while True:

    xold1 = x1
    xold2 = x2
    xold3 = x3
    count += 1
    x1 = ((b[0][0]) - (a[0][1]) * xold2 - (a[0][2]) * xold3) / a[0][0]
    x2 = ((b[1][0]) - (a[1][0]) * xold1 - (a[1][2]) * xold3) / a[1][1]
    x3 = ((b[2][0]) - (a[2][0]) * xold1 - (a[2][1]) * xold2) / a[2][2]

    if (((x1 - xold1) / x1) < es) and \
             ((x2-xold2)/x2 < es) and \
             ((x3-xold3)/x3 < es) :
        break


print('without relaxation')
print('x1 =', x1)
print('x2 =', x2)
print('x3 =', x3)
print()
print('Total used', count, "times.")

a = np.array([[ 2.0, -6.0, -1.0], [-3.0, -1.0, 7.0], [-8.0, 1.0, -2.0]]) b = np.array([[-38.0], [-34.0], [-20.0]]) def pivot(a, b): p = np.array((a), float) q = np.array((b), float) n = len(q) for i in range(0, n - 1): if abs(p[i, 1]) == 0: for k in range(i + 1, n): if abs((p[k, i])) > abs(p[i, i]): p[[i, k]] = p[[k, i]] q[[i, k]] = q[[k, i]] break for j in range(i + 1, n): f = p[j][i] / p[i][i] p[j, :] = p[j, :] - f * p[i, :] q[j] = q[j] -f * q[i] return p,q a, b = pivot(a, b) x1 = 1 x2 = 1 x3 = 1 count = 0 es = 0.05 while True: xold1 = x1 xold2 = x2 xold3 = x3 count += 1 x1 = ((b[0][0]) - (a[0][1]) * xold2 - (a[0][2]) * xold3) / a[0][0] x2 = ((b[1][0]) - (a[1][0]) * xold1 - (a[1][2]) * xold3) / a[1][1] x3 = ((b[2][0]) - (a[2][0]) * xold1 - (a[2][1]) * xold2) / a[2][2] if (((x1 - xold1) / x1) < es) and \ ((x2-xold2)/x2 < es) and \ ((x3-xold3)/x3 < es) : break print('without relaxation') print('x1 =', x1) print('x2 =', x2) print('x3 =', x3) print() print('Total used', count, "times.")

without relaxation
x1 = 4.0
x2 = 8.0
x3 = -1.9999999999999993

Total used 3 times.

2. with relaxation.¶

In [3]:

a, b = pivot(a, b)
x1 = 1
x2 = 1
x3 = 1
count = 0
c = 1.2
es = 0.05

while True:
    xold1 = x1
    xold2 = x2
    xold3 = x3
    count += 1
    x1 = ((b[0][0]) - (a[0][1]) * xold2 - (a[0][2]) * xold3) / a[0][0]
    x2 = ((b[1][0]) - (a[1][0]) * xold1 - (a[1][2]) * xold3) / a[1][1]
    x3 = ((b[2][0]) - (a[2][0]) * xold1 - (a[2][1]) * xold2) / a[2][2]
    x1 = c * x1 + (1 - c) * xold1
    x2 = c * x2 + (1 - c) * xold2
    x3 = c * x3 + (1 - c) * xold3

    if (((x1 - xold1) / x1) < es) and \
             ((x2-xold2)/x2 < es) and \
             ((x3-xold3)/x3 < es) :
        break
    

print('with relaxation')
print('x1=',x1)
print('x2=',x2)
print('x3=',x3)
print()
print('Total used', count, "times.")

a, b = pivot(a, b) x1 = 1 x2 = 1 x3 = 1 count = 0 c = 1.2 es = 0.05 while True: xold1 = x1 xold2 = x2 xold3 = x3 count += 1 x1 = ((b[0][0]) - (a[0][1]) * xold2 - (a[0][2]) * xold3) / a[0][0] x2 = ((b[1][0]) - (a[1][0]) * xold1 - (a[1][2]) * xold3) / a[1][1] x3 = ((b[2][0]) - (a[2][0]) * xold1 - (a[2][1]) * xold2) / a[2][2] x1 = c * x1 + (1 - c) * xold1 x2 = c * x2 + (1 - c) * xold2 x3 = c * x3 + (1 - c) * xold3 if (((x1 - xold1) / x1) < es) and \ ((x2-xold2)/x2 < es) and \ ((x3-xold3)/x3 < es) : break print('with relaxation') print('x1=',x1) print('x2=',x2) print('x3=',x3) print() print('Total used', count, "times.")

with relaxation
x1= 3.2435200000000086
x2= 8.018080000000001
x3= -2.000959999999999

Total used 5 times.

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Last update: 2024-04-27