Python学习：Numpy 和 Scipy 练习

Generate matrices A, withrandom Gaussian entries, B, a Toeplitz matrix, where A ∈ Rn×m and B ∈ Rm×m, for n = 200, m = 500. 

Exercise 9.1: Matrixoperations

Calculate A + A, AA⊤, A⊤A and AB. Write a function that computes A(B − λI) forany λ. 

输入样例

100

样例结果

input n 
100
[[ 0.25492905  0.23925104 -1.98625118 ...  3.12361183 -1.76108268
  -1.03045014]
 [ 0.99610086 -2.19372995  1.99450751 ...  1.62936487 -0.79844217
  -1.01313104]
 [ 1.02039602 -1.18865333  0.90970572 ... -0.00643825  0.37541486
  -2.6062939 ]
 ...
 [ 2.49890466 -0.03042841 -0.40941104 ...  2.83135408  1.56049787
  -1.75867723]
 [ 1.77857431  3.6898885  -1.06631876 ... -1.28950938 -3.34071184
  -0.20809047]
 [ 1.02772208 -1.27415573 -1.98211949 ... -2.66795933  0.94557247
   0.39738567]]
[[497.10157725  29.33997367  11.89869076 ... -39.2352458  -28.08448026
  -16.66887871]
 [ 29.33997367 516.0630948   10.41121829 ...  25.35670707 -16.97177249
  -23.11154664]
 [ 11.89869076  10.41121829 465.19591596 ... -36.1198177   -1.008844
  -27.82534697]
 ...
 [-39.2352458   25.35670707 -36.1198177  ... 482.91736413  20.5736587
  -14.02801916]
 [-28.08448026 -16.97177249  -1.008844   ...  20.5736587  557.17793535
   27.50731586]
 [-16.66887871 -23.11154664 -27.82534697 ... -14.02801916  27.50731586
  462.0287267 ]]
[[217.98650996  12.01549371  -1.81752272 ... -12.78807957   3.16064157
    8.67172505]
 [ 12.01549371 225.30981342   9.86966004 ...  -0.67555045  -1.59276503
    7.04849011]
 [ -1.81752272   9.86966004 176.43297755 ...  -5.20047595  14.58406069
   -3.65044271]
 ...
 [-12.78807957  -0.67555045  -5.20047595 ... 207.4008876   12.04331292
   15.13019973]
 [  3.16064157  -1.59276503  14.58406069 ...  12.04331292 161.20307071
   26.59891546]
 [  8.67172505   7.04849011  -3.65044271 ...  15.13019973  26.59891546
  209.15421597]]
[[-2968.75941953 -2951.42722365 -2933.85577673 ... -5556.61871045
  -5570.90444446 -5586.95126115]
 [ 2687.82633874  2680.12814943  2670.23623017 ...  1647.79975151
   1658.30561489  1668.01303611]
 [-3585.77473724 -3559.94286365 -3535.29964339 ... -8799.9897419
  -8822.57034043 -8844.7755241 ]
 ...
 [-2167.51025683 -2161.4846763  -2155.48952416 ...   405.74213491
    402.41363838   400.64563972]
 [-7925.52176622 -7896.19366971 -7863.17568469 ... -5825.44670401
  -5849.44742391 -5876.78885564]
 [ 9772.60466705  9735.48683747  9697.09485215 ...  9263.76595799
   9300.56855152  9338.31671751]]
[[-2981.5058721  -2963.38977575 -2834.54321794 ... -5712.79930194
  -5482.85031043 -5535.42875429]
 [ 2638.02129558  2789.81464714  2570.51085476 ...  1566.33150826
   1698.22772352  1718.66958831]
 [-3636.7945383  -3500.51019699 -3580.78492929 ... -8799.66782921
  -8841.34108365 -8714.46082924]
 ...
 [-2292.45548961 -2159.96325601 -2135.01897228 ...   264.17443092
    324.38874494   488.57950099]
 [-8014.4504819  -8080.68809487 -7809.85974688 ... -5760.97123503
  -5682.41183203 -5866.38433235]
 [ 9721.21856286  9799.19462415  9796.20082673 ...  9397.16392435
   9253.28992805  9318.44743384]]

代码实现

import numpy as np

A = np.random.randn(200,500)
B = np.zeros((500,500))
C = np.zeros((500,500))
n = input("input n \n")

for i in range(0,500):
	for j in range(0,500):
		if i==0:
			B[i][j]=j+1
		if j==0:
			B[i][j]=i+1
		else:
			if i>=j:
				B[i][j]=B[i-j][0]		
			else:
				B[i][j]=B[0][j-i]

for i in range(0,500):
	for j in range(0,500):
		if i==j:
			C[i][j]=n

res_1 = A + A
res_2 = np.dot(A,A.transpose())
res_3 = np.dot(A.transpose(),A)
res_4 = np.dot(A,B)
res_5 = np.dot(A,(B-C))

print(res_1)
print(res_2)
print(res_3)
print(res_4)
print(res_5)

Exercise 9.2: Solving a linear system

Generate a vector b with mentries and solve Bx = b.

样例结果

x= [[ 8.34559935e-02]
 [-1.84147550e-01]
 [-1.05216680e-01]
 [ 2.38800669e-01]
 [-8.28615179e-02]
 [ 3.61193104e-01]
 [-4.45423501e-01]
 [-1.35173495e-01]
 [ 6.09494125e-01]
 [-4.80077298e-01]
 [ 3.29950112e-01]
 [-4.14958640e-01]
 [ 3.16616561e-01]
 [-7.51664965e-02]
 [ 1.51916913e-01]
 [-4.03210726e-01]
 [ 8.92947689e-02]
 [ 5.26897948e-01]
 [-6.16379359e-01]
 [ 2.08842675e-01]
 [-1.07060824e-01]
 [ 2.17467580e-01]
 [-1.66095536e-02]
 [-2.55367202e-01]
 [ 2.06886917e-01]
 [ 3.53684786e-02]
 [ 2.27446762e-01]
 [-3.93807802e-01]
 [ 4.59129164e-02]
 [-6.72456530e-02]
 [ 5.25469225e-01]
 [-7.37078550e-01]
 [ 4.35221198e-01]
 [ 9.20155706e-02]
 [-6.94621527e-02]
 [-1.79338551e-01]
 [-1.27424314e-01]
 [ 1.16619655e-01]
 [ 1.74149962e-01]
 [ 4.54203642e-02]
 [-7.16545186e-02]
 [-4.31852899e-01]
 [ 7.34156101e-01]
 [-4.03685085e-01]
 [-1.18011844e-01]
 [ 1.50847021e-01]
 [-2.22778242e-01]
 [ 2.38117117e-01]
 [ 2.90521359e-01]
 [-5.82160008e-01]
 [ 2.38669341e-01]
 [ 3.31672850e-01]
 [-6.45536733e-01]
 [ 5.68920365e-01]
 [-3.23782171e-01]
 [ 1.57215032e-01]
 [ 1.27013666e-01]
 [-9.00145957e-02]
 [-2.29290467e-01]
 [ 2.22098850e-01]
 [-1.19433020e-01]
 [-1.91330397e-01]
 [ 3.58191134e-01]
 [-1.98690801e-01]
 [-2.69714817e-02]
 [ 4.04991269e-01]
 [-5.07232204e-01]
 [ 2.75826290e-01]
 [-1.94707577e-01]
 [ 2.81194028e-01]
 [-2.62985699e-02]
 [-3.11719924e-01]
 [-3.49547695e-02]
 [ 4.19862396e-01]
 [-2.72425910e-01]
 [-1.85272247e-02]
 [ 3.99369635e-01]
 [-5.39964024e-01]
 [ 3.56416325e-01]
 [-4.62717345e-01]
 [ 7.22801876e-01]
 [-7.65389295e-01]
 [ 7.72949540e-01]
 [-6.46372969e-01]
 [ 3.21245808e-01]
 [-1.59407527e-01]
 [ 1.99854992e-01]
 [-3.95641322e-02]
 [-3.37475819e-01]
 [ 5.15962135e-01]
 [-6.42357283e-01]
 [ 6.66217134e-01]
 [-5.64741175e-01]
 [ 3.26223884e-01]
 [ 2.70030684e-01]
 [-3.82309935e-01]
 [ 2.14708519e-01]
 [-1.89896840e-01]
 [-3.29821144e-01]
 [ 4.05792796e-01]
 [ 1.06086499e-01]
 [-1.31467740e-01]
 [-1.70459142e-01]
 [ 4.04866655e-01]
 [-3.62953563e-01]
 [ 4.43021167e-02]
 [ 4.11584772e-01]
 [-4.39571880e-01]
 [ 1.75695807e-01]
 [-1.13816943e-01]
 [ 1.77936937e-01]
 [-4.38126179e-01]
 [ 1.46660064e-01]
 [ 3.93872145e-01]
 [-8.27819909e-02]
 [-4.01875853e-01]
 [ 2.86679346e-01]
 [-2.36906349e-01]
 [ 5.62543362e-01]
 [-4.83403971e-01]
 [ 5.61321843e-02]
 [ 1.83067574e-01]
 [-2.29046358e-02]
 [-3.81521836e-01]
 [ 4.51834916e-01]
 [-3.86659474e-01]
 [ 2.69395756e-01]
 [-8.42283052e-02]
 [ 5.04553135e-02]
 [ 3.51851272e-01]
 [-7.26952959e-01]
 [ 5.83419718e-01]
 [-4.13391983e-01]
 [ 2.48918661e-01]
 [ 2.29856832e-01]
 [-3.02926772e-01]
 [-3.62752746e-01]
 [ 4.00435333e-01]
 [ 5.29232042e-02]
 [-1.22067175e-01]
 [ 9.09945531e-02]
 [-1.66264653e-01]
 [ 9.23163391e-02]
 [-6.81396456e-03]
 [ 4.14225755e-01]
 [-2.94523645e-01]
 [-3.88799007e-01]
 [ 4.62103445e-01]
 [-6.63221933e-02]
 [-1.19887025e-01]
 [ 1.02216061e-01]
 [-4.39030717e-01]
 [ 3.72177725e-01]
 [ 3.18769836e-01]
 [-4.88558130e-01]
 [ 9.02108609e-02]
 [ 1.27080832e-04]
 [ 3.60026869e-01]
 [-3.39144470e-01]
 [-7.52131660e-02]
 [ 2.32187303e-01]
 [-1.75593875e-01]
 [-1.03410914e-01]
 [ 3.88736022e-01]
 [-4.03879555e-01]
 [ 4.77356751e-01]
 [-3.01255981e-01]
 [-3.80579114e-01]
 [ 7.34248653e-01]
 [-2.25040021e-01]
 [-4.81945652e-01]
 [ 3.10681057e-01]
 [ 4.64571053e-01]
 [-6.28670824e-01]
 [ 2.21569530e-02]
 [ 2.15389878e-01]
 [ 1.93473475e-01]
 [-2.43842107e-01]
 [-1.37873571e-01]
 [ 1.20200249e-01]
 [ 1.16553973e-01]
 [-1.36284792e-01]
 [ 1.66330161e-01]
 [-1.11419831e-01]
 [ 1.92658197e-01]
 [ 6.57110486e-03]
 [-3.01866787e-01]
 [ 1.82473580e-01]
 [-2.64987495e-01]
 [ 3.45321160e-01]
 [-3.09240988e-01]
 [ 1.33960265e-01]
 [-7.81492375e-02]
 [ 2.15875840e-01]
 [ 2.86054122e-02]
 [-3.17512728e-01]
 [ 2.62134381e-01]
 [-1.97039252e-01]
 [ 1.19289634e-02]
 [ 2.75734203e-01]
 [-4.76397098e-02]
 [-2.79887432e-01]
 [ 3.61766050e-01]
 [-2.78526045e-01]
 [-1.72145444e-01]
 [ 6.27631305e-01]
 [-5.58907569e-01]
 [ 5.89293065e-02]
 [ 9.14401894e-02]
 [ 2.61608242e-01]
 [-2.66915108e-01]
 [-4.44390763e-02]
 [-1.23150806e-01]
 [ 2.22662818e-01]
 [ 6.32052460e-02]
 [-9.71216172e-03]
 [-8.79564343e-02]
 [-1.01944916e-01]
 [ 4.02967664e-01]
 [-6.47554377e-01]
 [ 6.38357924e-01]
 [-3.05374054e-01]
 [-1.38076979e-01]
 [ 1.05168498e-01]
 [ 3.97267617e-01]
 [-3.96264915e-01]
 [ 4.48281181e-02]
 [-2.22862596e-01]
 [ 1.09344182e-01]
 [ 4.43018978e-01]
 [-4.64679415e-01]
 [ 1.42262443e-01]
 [ 1.10327577e-01]
 [-2.64022453e-01]
 [ 3.27853659e-01]
 [-1.93284004e-01]
 [ 1.55274242e-01]
 [-2.09814130e-01]
 [-2.13672781e-01]
 [ 4.24597156e-01]
 [ 8.08912195e-02]
 [-1.71812265e-01]
 [-2.89931707e-01]
 [ 4.63749854e-01]
 [-1.25509575e-01]
 [-2.72193729e-01]
 [-2.68369069e-02]
 [ 2.65616649e-01]
 [-9.83989391e-02]
 [ 9.46798612e-02]
 [-2.59852153e-02]
 [-1.49430526e-01]
 [ 1.16118304e-01]
 [ 1.93273221e-01]
 [-1.48173003e-01]
 [ 9.36635114e-02]
 [-3.83976164e-01]
 [ 2.46498955e-01]
 [ 1.53478280e-01]
 [-4.03415337e-01]
 [ 2.76743035e-01]
 [ 2.22226291e-01]
 [-1.64240752e-01]
 [-4.40253163e-01]
 [ 3.99668852e-01]
 [ 4.82014754e-02]
 [ 1.75574862e-01]
 [-2.30672219e-01]
 [-3.24456787e-02]
 [-3.62904835e-01]
 [ 6.13400317e-01]
 [-2.20835151e-01]
 [-5.44488314e-02]
 [ 1.46665363e-02]
 [-1.42919263e-02]
 [ 9.26625144e-02]
 [-1.39764451e-01]
 [-1.81767131e-01]
 [ 3.56835961e-01]
 [-1.36186882e-01]
 [ 1.08827016e-01]
 [-6.52437560e-02]
 [-3.16979832e-02]
 [-8.51227877e-02]
 [ 2.78903330e-01]
 [-8.93996755e-02]
 [-3.94409092e-01]
 [ 6.96196117e-01]
 [-8.40432688e-01]
 [ 5.62904334e-01]
 [-1.77707836e-01]
 [ 2.74438431e-01]
 [-9.79662880e-02]
 [-2.23726436e-01]
 [-1.13204908e-02]
 [ 6.00267052e-02]
 [ 3.60568231e-02]
 [ 7.75294951e-03]
 [ 2.67711629e-01]
 [-3.92684854e-01]
 [ 3.94335654e-01]
 [-1.19287430e-01]
 [-2.47033916e-01]
 [ 1.76772570e-01]
 [-4.51513879e-01]
 [ 5.56585270e-01]
 [-1.77922158e-01]
 [-3.82020340e-02]
 [ 2.56232688e-02]
 [ 2.00051703e-01]
 [ 4.13440867e-02]
 [-5.12434683e-01]
 [ 4.88874944e-01]
 [-3.62580327e-01]
 [ 1.88331626e-01]
 [ 7.44348311e-02]
 [-3.21932825e-01]
 [ 2.04789946e-01]
 [ 2.17712514e-01]
 [-4.55148413e-02]
 [-7.92044100e-03]
 [-5.92181532e-01]
 [ 8.65385461e-01]
 [-6.73814779e-01]
 [ 4.37877425e-01]
 [-2.87210111e-01]
 [ 3.10865718e-02]
 [ 2.70632243e-01]
 [-2.26927803e-01]
 [ 1.22344876e-02]
 [-6.99719097e-02]
 [-7.68293708e-02]
 [ 6.41160471e-01]
 [-8.86151645e-01]
 [ 6.05917014e-01]
 [-3.28296444e-01]
 [ 4.13690925e-01]
 [-2.18803165e-01]
 [-4.59454186e-01]
 [ 6.93952754e-01]
 [-3.00239616e-01]
 [-6.95195176e-02]
 [ 1.60753641e-01]
 [-1.36383513e-01]
 [ 1.14609218e-01]
 [-2.13078709e-02]
 [-3.04207299e-01]
 [ 2.21108764e-01]
 [ 4.91115735e-01]
 [-8.77625645e-01]
 [ 7.95131045e-01]
 [-3.92170945e-01]
 [ 2.57796756e-02]
 [ 9.78281444e-03]
 [-1.05304649e-01]
 [ 2.80584204e-01]
 [-2.71143030e-01]
 [ 1.33212841e-01]
 [-1.20549891e-02]
 [-3.19682269e-01]
 [ 5.23647780e-01]
 [-4.50021216e-01]
 [ 2.45933292e-01]
 [ 2.85081395e-01]
 [-4.63970297e-01]
 [-6.31085095e-02]
 [ 2.81840540e-01]
 [ 2.24538411e-01]
 [-4.88299093e-01]
 [-9.00847927e-02]
 [ 4.46814350e-01]
 [-2.04349858e-01]
 [ 2.83225413e-01]
 [-2.72634941e-01]
 [-1.94632537e-01]
 [ 5.15796396e-01]
 [-5.74948370e-01]
 [ 6.39920473e-01]
 [-3.53756033e-01]
 [-4.27445043e-01]
 [ 8.05442795e-01]
 [-4.46529405e-01]
 [-1.89904406e-01]
 [ 3.50387918e-01]
 [-2.14339396e-01]
 [ 3.16186957e-01]
 [-4.03410379e-01]
 [ 4.76154853e-01]
 [-4.64421972e-01]
 [ 1.83292581e-01]
 [ 3.22499713e-01]
 [-5.17954578e-01]
 [ 2.57497437e-02]
 [ 4.26516903e-01]
 [-2.93560668e-01]
 [ 5.34450834e-02]
 [-1.07426566e-01]
 [ 3.65050492e-01]
 [-2.44835758e-01]
 [-1.22053106e-01]
 [ 6.80592474e-02]
 [-1.12267718e-01]
 [ 3.73459765e-01]
 [-1.60091991e-01]
 [-3.44004770e-01]
 [ 5.82848684e-01]
 [-3.32855543e-01]
 [-4.97010589e-03]
 [-2.67571979e-01]
 [ 5.17533714e-01]
 [-3.23625173e-01]
 [ 4.15842209e-02]
 [ 3.09994186e-01]
 [-2.76155644e-01]
 [ 1.59719627e-01]
 [-5.54390588e-01]
 [ 6.29207519e-01]
 [-2.93647246e-01]
 [-4.47903271e-03]
 [ 3.30323722e-01]
 [-2.98414529e-01]
 [ 1.30068947e-01]
 [-7.14049009e-02]
 [ 1.58201781e-01]
 [-1.38722315e-01]
 [-8.60606599e-02]
 [ 1.05736489e-01]
 [-5.61986036e-02]
 [ 5.48823240e-03]
 [-3.24105544e-03]
 [ 1.83805758e-01]
 [-5.87915789e-01]
 [ 6.69007531e-01]
 [-2.50181699e-01]
 [-1.84397493e-01]
 [ 2.62083489e-01]
 [-1.45250215e-01]
 [ 1.02866927e-01]
 [-7.57170519e-02]
 [ 1.78233776e-01]
 [-9.05827261e-02]
 [-6.82608355e-02]
 [-1.32277473e-01]
 [ 3.30129118e-01]
 [-1.07163280e-01]
 [-9.70541969e-02]
 [ 2.40832779e-01]
 [-2.07909789e-01]
 [-1.83596140e-01]
 [ 3.64407398e-01]
 [-1.58821627e-01]
 [-9.74028243e-02]
 [ 8.14949093e-03]
 [-2.05600906e-01]
 [ 3.97200003e-01]
 [-8.12137115e-02]
 [-1.88594692e-02]
 [ 1.93518228e-02]
 [ 4.08701301e-02]
 [-1.64708739e-01]
 [ 1.32983868e-01]
 [-3.63054775e-01]
 [ 4.45069156e-01]
 [-2.90423043e-01]
 [ 3.24980432e-01]
 [-3.19020489e-01]
 [ 3.14806943e-01]
 [-3.57321337e-01]
 [ 5.41230863e-01]
 [-4.10219821e-01]
 [ 1.52549049e-01]
 [-2.07424603e-01]
 [ 9.33934478e-03]
 [ 3.44196720e-01]
 [-4.13172854e-01]
 [ 3.30351892e-01]
 [-2.39256653e-01]
 [-3.56589209e-03]
 [ 2.32317065e-01]
 [-1.13635358e-02]
 [-5.02270524e-01]
 [ 3.65358406e-01]
 [ 1.86624380e-01]
 [-1.60279199e-01]
 [ 1.58237224e-01]
 [-4.29898678e-01]
 [ 1.29692092e-01]
 [ 2.78657020e-01]
 [ 1.97796122e-02]
 [-2.98963634e-01]
 [ 1.23725825e-01]
 [-3.13556894e-01]
 [ 3.67464513e-01]
 [ 2.61752766e-01]
 [-3.70946403e-01]
 [-1.59506372e-01]
 [ 2.97173904e-01]
 [ 8.95554286e-02]
 [-4.57052916e-01]
 [ 2.83672682e-01]]

代码实现

import numpy as np

A = np.random.randn(200,500)
B = np.zeros((500,500))
C = []
for i in range(0,500):
	C.append(i)
b = np.random.rand(1,500)
b = b.transpose()

for i in range(0,500):
	for j in range(0,500):
		if i==0:
			B[i][j]=j+1
		if j==0:
			B[i][j]=i+1
		else:
			if i>=j:
				B[i][j]=B[i-j][0]		
			else:
				B[i][j]=B[0][j-i]

x=np.linalg.solve(B,b)
print("x= ",end="")
print(x)

Exercise 9.3: Norms

Compute the Frobenius norm ofA: ∥A∥F and the infinity norm of B: ∥B∥∞. Also find the largest and smallest singular values ofB.

样例结果

The the Frobenius norm of A:316.1542742333776
the infinity norm of B: 125250.0
The largest singular value of B: 87334.52045641872
The smallest singular value of B: 0.5000049348346122

代码实现

import numpy as np

A = np.random.randn(200,500)
B = np.zeros((500,500))
C = np.zeros((500,500))

for i in range(0,500):
	for j in range(0,500):
		if i==0:
			B[i][j]=j+1
		if j==0:
			B[i][j]=i+1
		else:
			if i>=j:
				B[i][j]=B[i-j][0]		
			else:
				B[i][j]=B[0][j-i]

U,Simga,VT = np.linalg.svd(B)

print("The the Frobenius norm of A:", end="")
print(np.linalg.norm(A,ord="fro") )
print("the infinity norm of B: ", end="")
print(np.linalg.norm(B,ord=np.inf) )
print("The largest singular value of B: ", end="")
print(max(Simga))
print("The smallest singular value of B: ", end="")
print(min(Simga))

Exercise 9.4: Power iteration

Generate a matrix Z, n × n,with Gaussian entries, and use the power iteration to find the largesteigenvalue and corresponding eigenvector of Z. How many iterations are neededtill convergence?

Optional: use the time.clock()method to compare computation time when varying n.

样例结果

corresponding eigenvector: 
[[0.91559639]
 [0.89268121]
 [0.87144762]
 [0.87458469]
 [0.93544372]
 [0.92239593]
 [0.9580073 ]
 [0.85998745]
 [0.86710414]
 [0.89849702]
 [0.9327396 ]
 [0.92985359]
 [0.92492912]
 [0.94828568]
 [0.87524395]
 [0.9571552 ]
 [0.83977028]
 [0.9171816 ]
 [0.97666198]
 [0.90403646]
 [0.88322289]
 [0.9610705 ]
 [0.89764944]
 [0.91319393]
 [1.        ]
 [0.84704578]
 [0.85399273]
 [0.9724098 ]
 [0.86084083]
 [0.9595304 ]
 [0.79744201]
 [0.92509031]
 [0.90543023]
 [0.81592596]
 [0.87793256]
 [0.79259957]
 [0.95003318]
 [0.89409636]
 [0.89796365]
 [0.84981036]
 [0.93887177]
 [0.92832027]
 [0.89465133]
 [0.9442769 ]
 [0.8596261 ]
 [0.90192049]
 [0.9129606 ]
 [0.90048482]
 [0.9391955 ]
 [0.91988383]
 [0.90919921]
 [0.93360468]
 [0.95790932]
 [0.90077368]
 [0.90525969]
 [0.94378787]
 [0.87498242]
 [0.88034921]
 [0.83672044]
 [0.95834085]
 [0.87479429]
 [0.96338713]
 [0.89796626]
 [0.94590632]
 [0.89370282]
 [0.87129726]
 [0.91268794]
 [0.91096478]
 [0.88757922]
 [0.87460715]
 [0.88463397]
 [0.85673907]
 [0.92399461]
 [0.91177149]
 [0.88993293]
 [0.85145739]
 [0.97344888]
 [0.93499517]
 [0.91170027]
 [0.94849799]
 [0.92809657]
 [0.85780627]
 [0.91054134]
 [0.90489654]
 [0.9329966 ]
 [0.87025115]
 [0.94732186]
 [0.85818768]
 [0.92345759]
 [0.90537044]
 [0.93544906]
 [0.96247193]
 [0.94489962]
 [0.83528039]
 [0.92233327]
 [0.9095816 ]
 [0.90990527]
 [0.93047281]
 [0.86281859]
 [0.86985438]
 [0.97422147]
 [0.89567786]
 [0.95886992]
 [0.89747697]
 [0.90123251]
 [0.93617358]
 [0.96096396]
 [0.87865638]
 [0.87203685]
 [0.91988447]
 [0.8323721 ]
 [0.95160632]
 [0.92246841]
 [0.90499615]
 [0.87873996]
 [0.93696255]
 [0.93369107]
 [0.87042756]
 [0.9077112 ]
 [0.88380723]
 [0.9604506 ]
 [0.97583191]
 [0.91643467]
 [0.89307383]
 [0.87370851]
 [0.9892885 ]
 [0.84000359]
 [0.85144256]
 [0.89242572]
 [0.83290574]
 [0.85820479]
 [0.87453674]
 [0.87551866]
 [0.8993992 ]
 [0.88702429]
 [0.94640939]
 [0.9016169 ]
 [0.93494489]
 [0.91671573]
 [0.96857068]
 [0.8708158 ]
 [0.93530149]
 [0.89689452]
 [0.95541044]
 [0.93012103]
 [0.91541332]
 [0.88572612]
 [0.88889341]
 [0.84604597]
 [0.91236831]
 [0.90697303]
 [0.85590704]
 [0.92512585]
 [0.95215093]
 [0.94143391]
 [0.981321  ]
 [0.91820463]
 [0.89896148]
 [0.91086301]
 [0.88439336]
 [0.91817239]
 [0.88880591]
 [0.89581724]
 [0.84920096]
 [0.86460557]
 [0.89611826]
 [0.84486929]
 [0.96190026]
 [0.96311036]
 [0.90073018]
 [0.89755858]
 [0.90938207]
 [0.88700873]
 [0.9257148 ]
 [0.85543779]
 [0.90367665]
 [0.91201713]
 [0.87187459]
 [0.94117184]
 [0.90071036]
 [0.83150206]
 [0.96937621]
 [0.92196265]
 [0.92568097]
 [0.82051319]
 [0.93548524]
 [0.92170302]
 [0.91919326]
 [0.89899595]
 [0.90106145]
 [0.93684119]
 [0.95047289]
 [0.92568223]
 [0.84363326]
 [0.96187736]
 [0.91418153]
 [0.97973243]
 [0.88874558]
 [0.97534505]
 [0.87660444]]
eigenvalue: 100.52496392461119
iterations: 6
Time: 0.0003459999999999852

代码实现

import numpy as np
import time
import scipy.linalg
import scipy

Z = np.mat(np.random.rand(200,200))
v = np.mat(np.ones(200)) #初始向量
n = 0	#迭代次数
v = v.T
x = v
t_ = 0 
ei = scipy.linalg.norm(x,ord=np.inf)  #特征值

time_ = time.clock()

while np.abs(t_-ei)>1e-4:
	t_ = ei
	x = Z * v
	ei = scipy.linalg.norm(x,ord=np.inf)
	v = x/ei
	n = n+1

time_ = time.clock()-time_

print("corresponding eigenvector: ")
print(v)
print("eigenvalue: ",end="")
print(ei)
print("iterations: ",end="")
print(n)
print("Time: ",end="")
print(time_)

Exercise 9.5: Singular values

Generate an n × n matrix,denoted by C, where each entry is 1 with probability p and 0 otherwise. Use thelinear algebra library of Scipy to compute the singular values of C. What canyou say about the relationship between n, p and the largest singular value?

样例结果

p: 0.8
n: 70
the largest singular value of C: 56.197368953325764
p: 0.5
n: 20
the largest singular value of C: 10.779133254881227
p: 0.1
n: 30
the largest singular value of C: 3.9385153803168302
p: 0.4
n: 20
the largest singular value of C: 8.82997237316175
p: 0.5
n: 50
the largest singular value of C: 25.861788207081943
p: 0.5
n: 20
the largest singular value of C: 10.85155130553588
p: 0.9
n: 50
the largest singular value of C: 44.98491751547517
p: 0.1
n: 80
the largest singular value of C: 9.134182074856593
p: 0.2
n: 90
the largest singular value of C: 18.711843187908286
p: 0.8
n: 40
the largest singular value of C: 31.725598579547427

代码实现

import numpy as np
import scipy.linalg as la
import random

for i in range(0,10):
	p=random.randint(1,9)/10
	n=random.randint(1,9)*10
	C=la.svdvals(np.random.binomial(1, p, [n,n]))
	print("p: ", end="")
	print(p)
	print("n: ", end= "")
	print(n)
	print("the largest singular value of C: ", end="")
	print(max(C))

Exercise 9.6: Nearest neighbor

Write a function that takes avalue z and an array A and finds the element in A that is closest to z. Thefunction should return the closest value, not index.

Hint: Use the built-infunctionality of Numpy rather than writing code to find this value manually. Inparticular, use brackets and argmin.

样例结果

random z: 0.885393383649773
cloest value: 0.8853989213942045

代码实现

import numpy as np
import scipy

A = np.random.randn(200,500)
A = A.reshape(1,100000)
z = np.random.random()

def f(a, z):
	c = np.ones(100000)
	c = z*c 
	return a[0,np.argmin(np.fabs(c-a))]

re =f(A,z)


print("random z: ", end = "")
print(z)

print("cloest value: ", end="")
print(re)