Solving linear equations by different methods
Contents
Solving linear equations by different methods¶
Randall Romero Aguilar, PhD
This demo is based on the original Matlab demo accompanying the Computational Economics and Finance 2001 textbook by Mario Miranda and Paul Fackler.
Original (Matlab) CompEcon file: demlin01.m
Running this file requires the Python version of CompEcon. This can be installed with pip by running
!pip install compecon --upgrade
Last updated: 2022-Ago-07
import numpy as np
import pandas as pd
from numpy.linalg import solve, inv
from timeit import default_timer as timer
from numba import jit
Make a function to time
tic = lambda: timer()
toc = lambda t: 1000* (timer() - t) # ellapsed milliseconds
Milliseconds required to solve n by n linear equation \(Ax = b\)¶
m times using solve(A, b) and dot(inv(A), b), computing inverse only once.
mvalues = [1, 100]
nvalues = [50, 500]
cases = pd.MultiIndex.from_product([mvalues, nvalues], names=['m','n'])
results0 = pd.DataFrame(index=cases, columns=['solve(A,b)', 'inv(A) @ b'])
for m, n in cases:
A = np.random.rand(n, n)
b = np.random.rand(n, 1)
tt = tic()
for j in range(m):
x = solve(A, b)
results0.loc[(m, n), 'solve(A,b)'] = toc(tt)
tt = tic()
Ainv = inv(A)
for j in range(m):
x = Ainv @ b
results0.loc[(m, n), 'inv(A) @ b'] = toc(tt)
@jit
def using_solve(A, b, m):
for j in range(m):
x = solve(A, b)
@jit
def using_inv(Ainv, b, m):
for j in range(m):
x = Ainv @ b
#run once to compile
using_solve(A, b, m)
using_inv(Ainv, b, m)
results1 = pd.DataFrame(index=cases, columns=['solve(A,b)', 'inv(A) @ b'])
for m, n in cases:
A = np.random.rand(n, n)
b = np.random.rand(n, 1)
tt = tic()
using_solve(A, b, m)
results1.loc[(m, n), 'solve(A,b)'] = toc(tt)
tt = tic()
Ainv = inv(A)
using_inv(Ainv, b, m)
results1.loc[(m, n), 'inv(A) @ b'] = toc(tt)
pd.concat([results0, results1], keys=['without jit', 'using jit'], axis=1).style.highlight_min(axis=1)
| without jit | using jit | ||||
|---|---|---|---|---|---|
| solve(A,b) | inv(A) @ b | solve(A,b) | inv(A) @ b | ||
| m | n | ||||
| 1 | 50 | 7.494000 | 0.731900 | 0.083900 | 0.309500 |
| 500 | 3.313000 | 9.448000 | 3.277700 | 8.304200 | |
| 100 | 50 | 7.089400 | 0.838400 | 4.628700 | 0.391200 |
| 500 | 777.569700 | 19.509600 | 345.602300 | 13.260500 | |