python scipy庫

3、假定正態分佈，求解1倍標準差和0.5倍標準差的機率？

2、求解多元線性或非線性方程組解

1、求解3元一次方程

 

一、學習資料  https://github.com/lijin-THU/notes-python/tree/master/04-scipy

二、子模塊，即功能

三、學習筆記

2、help(linalg)

dir(linalg)
['absolute_import',
 'basic',
 'blas',
 'block_diag',
 'cdf2rdf',
 'cho_factor',
 'cho_solve',
 'cho_solve_banded',
 'cholesky',
 'cholesky_banded',
 'circulant',
 'clarkson_woodruff_transform',
 'companion',
 'coshm',
 'cosm',
 'cython_blas',
 'cython_lapack',
 'decomp',
 'decomp_cholesky',
 'decomp_lu',
 'decomp_qr',
 'decomp_schur',
 'decomp_svd',
 'det',
 'dft',
 'diagsvd',
 'division',
 'eig',
 'eig_banded',
 'eigh',
 'eigh_tridiagonal',
 'eigvals',
 'eigvals_banded',
 'eigvalsh',
 'eigvalsh_tridiagonal',
 'expm',
 'expm_cond',
 'expm_frechet',
 'find_best_blas_type',
 'flinalg',
 'fractional_matrix_power',
 'funm',
 'get_blas_funcs',
 'get_lapack_funcs',
 'hadamard',
 'hankel',
 'helmert',
 'hessenberg',
 'hilbert',
 'inv',
 'invhilbert',
 'invpascal',
 'kron',
 'lapack',
 'ldl',
 'leslie',
 'linalg_version',
 'logm',
 'lstsq',
 'lu',
 'lu_factor',
 'lu_solve',
 'matfuncs',
 'matrix_balance',
 'misc',
 'norm',
 'null_space',
 'ordqz',
 'orth',
 'orthogonal_procrustes',
 'pascal',
 'pinv',
 'pinv2',
 'pinvh',
 'polar',
 'print_function',
 'qr',
 'qr_delete',
 'qr_insert',
 'qr_multiply',
 'qr_update',
 'qz',
 'rq',
 'rsf2csf',
 'schur',
 'signm',
 'sinhm',
 'sinm',
 'solve',
 'solve_banded',
 'solve_circulant',
 'solve_continuous_are',
 'solve_continuous_lyapunov',
 'solve_discrete_are',
 'solve_discrete_lyapunov',
 'solve_lyapunov',
 'solve_sylvester',
 'solve_toeplitz',
 'solve_triangular',
 'solveh_banded',
 'special_matrices',
 'sqrtm',
 'subspace_angles',
 'svd',
 'svdvals',
 'tanhm',
 'tanm',
 'test',
 'toeplitz',
 'tri',
 'tril',
 'triu']

 dir(optimize)

[ 'absolute_import', 'anderson', 'approx_fprime', 'basinhopping', 'bisect', 'bracket', 'brent', 'brenth', 'brentq', 'broyden1', 'broyden2', 'brute', 'check_grad', 'cobyla', 'curve_fit', 'diagbroyden', 'differential_evolution', 'division', 'excitingmixing', 'fixed_point', 'fmin', 'fmin_bfgs', 'fmin_cg', 'fmin_cobyla', 'fmin_l_bfgs_b', 'fmin_ncg', 'fmin_powell', 'fmin_slsqp', 'fmin_tnc', 'fminbound', 'fsolve', 'golden', 'lbfgsb', 'least_squares', 'leastsq', 'line_search', 'linear_sum_assignment', 'linearmixing', 'linesearch', 'linprog', 'linprog_verbose_callback', 'lsq_linear', 'minimize', 'minimize_scalar', 'minpack', 'minpack2', 'moduleTNC', 'newton', 'newton_krylov', 'nnls', 'nonlin', 'optimize', 'print_function', 'ridder', 'root', 'rosen', 'rosen_der', 'rosen_hess', 'rosen_hess_prod', 'show_options', 'slsqp', 'test', 'tnc', 'zeros']

問題：

一、scipy scipy-ref-1.1.0.pdf 中 Unconstrained minimization of multivariate scalar functions 下面

Nelder-Mead Simplex algorithm (method='Nelder-Mead')

不明白這個函數是如何求解的？爲啥要這樣寫？

def rosen(x):
    return sum(100.0*(x[1:]-x[:-1]**2.0)**2.0+(1-x[:-1])**2.0)

 

二、如何處理非線性約束問題？

1、若是所有爲線性約束問題，全部的線性約束問題均可以轉換爲矩陣來求解

例如：求解 x-y>0 y>2 線性約束下的 min(x)

2、僅存在一個非線性約束
3、存在多個非線性約束問題
4、同時存在多個非線性約束和線性約束問題

三、 在多元非線性約束下，爲何加入Jacobian和Hessians ?jacobian爲非線性約束的導數，Hessians這個函數是如何得出的？

 

3、假定正態分佈，求解1倍標準差和0.5倍標準差的機率？

https://baike.baidu.com/item/%E6%A0%87%E5%87%86%E5%B7%AE/1415772?fr=aladdin

import scipy.stats
1-scipy.stats.norm(0,1).cdf(1)
Out[3]: 0.15865525393145707
scipy.stats.norm(0,1).cdf(1)
Out[4]: 0.8413447460685429
1-(1-scipy.stats.norm(0,1).cdf(1))*2
Out[5]: 0.6826894921370859
1-(1-scipy.stats.norm(0,1).cdf(2))*2
Out[6]: 0.9544997361036416

 

1、求解3元一次方程

from scipy import linalg
A=np.array([[1,3,5],[2,5,1],[2,3,8]])
b=np.array([10,8,3])

for i in range(1000):
    x=linalg.solve(A,b)
    
x
Out[19]: array([-9.28,  5.16,  0.76])

 

2、求解多元線性或非線性方程組解

問題：只能獲得一個解，不能獲得所有解

from scipy.integrate import odeint
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import root,fsolve
#plt.rc('text', usetex=True) #使用latex
## 使用scipy.optimize模塊的root和fsolve函數進行數值求解方程
## 一、求解f(x)=2*sin(x)-x+1
# rangex1 = np.linspace(-2,8)
# rangey1_1,rangey1_2 = 2*np.sin(rangex1),rangex1-1
# plt.figure(1)
# plt.plot(rangex1,rangey1_1,'r',rangex1,rangey1_2,'b--')
# plt.title('$2sin(x)$ and $x-1$')
f1=lambda x:np.sin(x)*2-x+1
sol1_root = root(f1,[0])
print('sol1_root:',sol1_root)
print('sol1_root.x',sol1_root.x)
sol1_fsolve = fsolve(f1,[0])
print('sol1_fsolve:',sol1_fsolve)
print('----------------')
# 二、求解線性方程組{3X1+2X2=3;X1-2X2=5}
def f2(x):
    return np.array([3*x[0]+2*x[1]-3,x[0]-2*x[1]-5])
f2=lambda x:np.array([3*x[0]+2*x[1]-3,x[0]-2*x[1]-5])
sol2_root = root(f2,[0,0])
sol2_fsolve = fsolve(f2,[0,0])
print('sol2_fsolve:',sol2_fsolve) # [2. -1.5]
a = np.array([[3,2],[1,-2]])
b = np.array([3,5])
x = np.linalg.solve(a,b)
print('x:',x) # [2. -1.5]
## 三、求解非線性方程組
def f3(x):
    return np.array([2*x[0]**2+3*x[1]-3*x[2]**3-7,
                    x[0]+4*x[1]**2+8*x[2]-10,
                    x[0]-2*x[1]**3-2*x[2]**2+1])
sol3_root = root(f3,[0,0,0])
sol3_fsolve = fsolve(f3,[0,0,0])
print('sol3_fsolve:',sol3_fsolve)
Backend Qt5Agg is interactive backend. Turning interactive mode on.
sol1_root:     fjac: array([[-1.]])
     fun: array([0.31514905])
 message: 'The iteration is not making good progress, as measured by the \n  improvement from the last ten iterations.'
    nfev: 24
     qtf: array([-0.31514905])
       r: array([0.00451924])
  status: 5
 success: False
       x: array([-1.04882813])
sol1_root.x [-1.04882813]
sol1_fsolve: [-1.04882813]
sol2_fsolve: [ 2.  -1.5]
x: [ 2.  -1.5]
sol3_fsolve: [1.52964909 0.973546   0.58489796]