最近开始学习UFLDL，一个deep learning的教程
Exercise:Sparse Autoencoder
http://ufldl.stanford.edu/wiki/index.php/Exercise:Sparse_Autoencoder
The implemetantion needs to fill the blank code in those three files:sampleIMAGES.m, sparseAutoencoderCost.m, computeNumericalGradient.m.
There is one problem that the process has exceeded maximum number of iterations. I am not sure it is right or wrong. It seems we got the right training result so that I pasted all the code here. It used around 15 minutes to get the final result.
We randomly choose 10000 patches from 10 images. The training dataset looks like below:
The result visualization we obtained:
The source code
sampleIMAGES.m
function patches = sampleIMAGES()
% sampleIMAGES
% Returns 10000 patches for training
load IMAGES; % load images from disk
patchsize = 8; % we'll use 8x8 patches
numpatches = 10000;
% Initialize patches with zeros. Your code will fill in this matrix--one
% column per patch, 10000 columns.
patches = zeros(patchsize*patchsize, numpatches);
%% ---------- YOUR CODE HERE --------------------------------------
% Instructions: Fill in the variable called "patches" using data
% from IMAGES.
%
% IMAGES is a 3D array containing 10 images
% For instance, IMAGES(:,:,6) is a 512x512 array containing the 6th image,
% and you can type "imagesc(IMAGES(:,:,6)), colormap gray;" to visualize
% it. (The contrast on these images look a bit off because they have
% been preprocessed using using "whitening." See the lecture notes for
% more details.) As a second example, IMAGES(21:30,21:30,1) is an image
% patch corresponding to the pixels in the block (21,21) to (30,30) of
% Image 1
for num = 1:numpatches
index = ceil(10*rand);
[rowSize colSize] = size(IMAGES(:,:,index));
row = ceil((rowSize-patchsize+1)*rand);
col = ceil((colSize-patchsize+1)*rand);
imgv = reshape(IMAGES(row:row+7,col:col+7,index),64,1);
patches(:,num) = imgv;
end
%% ---------------------------------------------------------------
% For the autoencoder to work well we need to normalize the data
% Specifically, since the output of the network is bounded between [0,1]
% (due to the sigmoid activation function), we have to make sure
% the range of pixel values is also bounded between [0,1]
patches = normalizeData(patches);
end
%% ---------------------------------------------------------------
function patches = normalizeData(patches)
% Squash data to [0.1, 0.9] since we use sigmoid as the activation
% function in the output layer
% Remove DC (mean of images).
patches = bsxfun(@minus, patches, mean(patches));
% Truncate to +/-3 standard deviations and scale to -1 to 1
pstd = 3 * std(patches(:));
patches = max(min(patches, pstd), -pstd) / pstd;
% Rescale from [-1,1] to [0.1,0.9]
patches = (patches + 1) * 0.4 + 0.1;
end
sparseAutoencoderCost.m
function [cost,grad] = sparseAutoencoderCost(theta, visibleSize, hiddenSize, ...
lambda, sparsityParam, beta, data)
% visibleSize: the number of input units (probably 64)
% hiddenSize: the number of hidden units (probably 25)
% lambda: weight decay parameter
% sparsityParam: The desired average activation for the hidden units (denoted in the lecture
% notes by the greek alphabet rho, which looks like a lower-case "p").
% beta: weight of sparsity penalty term
% data: Our 64x10000 matrix containing the training data. So, data(:,i) is the i-th training example.
% The input theta is a vector (because minFunc expects the parameters to be a vector).
% We first convert theta to the (W1, W2, b1, b2) matrix/vector format, so that this
% follows the notation convention of the lecture notes.
W1 = reshape(theta(1:hiddenSize*visibleSize), hiddenSize, visibleSize);
W2 = reshape(theta(hiddenSize*visibleSize+1:2*hiddenSize*visibleSize), visibleSize, hiddenSize);
b1 = theta(2*hiddenSize*visibleSize+1:2*hiddenSize*visibleSize+hiddenSize);
b2 = theta(2*hiddenSize*visibleSize+hiddenSize+1:end);
% Cost and gradient variables (your code needs to compute these values).
% Here, we initialize them to zeros.
cost = 0;
W1grad = zeros(size(W1));
W2grad = zeros(size(W2));
b1grad = zeros(size(b1));
b2grad = zeros(size(b2));
%% ---------- YOUR CODE HERE --------------------------------------
% Instructions: Compute the cost/optimization objective J_sparse(W,b) for the Sparse Autoencoder,
% and the corresponding gradients W1grad, W2grad, b1grad, b2grad.
%
% W1grad, W2grad, b1grad and b2grad should be computed using backpropagation.
% Note that W1grad has the same dimensions as W1, b1grad has the same dimensions
% as b1, etc. Your code should set W1grad to be the partial derivative of J_sparse(W,b) with
% respect to W1. I.e., W1grad(i,j) should be the partial derivative of J_sparse(W,b)
% with respect to the input parameter W1(i,j). Thus, W1grad should be equal to the term
% [(1/m) \Delta W^{(1)} + \lambda W^{(1)}] in the last block of pseudo-code in Section 2.2
% of the lecture notes (and similarly for W2grad, b1grad, b2grad).
%
% Stated differently, if we were using batch gradient descent to optimize the parameters,
% the gradient descent update to W1 would be W1 := W1 - alpha * W1grad, and similarly for W2, b1, b2.
%
JWb = 0; %squared-error cost
JsparesWb = 0; %overall cost function:JsparesWb = JWb + beta*KL
[n m] = size(data); %m:sample number
%layer 2
z2 = W1*data + repmat(b1,1,m);
a2 = sigmoid(z2);
%layer 3
z3 = W2*a2 + repmat(b2,1,m);
a3 = sigmoid(z3);
%cost function J(W,b):average sum-of-squares error
sumofsquares = (0.5/m)*(sum(sum((a3-data).^2)));
JWb = sumofsquares + 0.5*lambda*(sum(sum(W1.^2)) + sum(sum(W2.^2)));
%sparsity parameter
%average activation in layer 2
rho = (1/m)*(sum(a2,2));
KL = sparsityParam .* log(sparsityParam./rho) + (1-sparsityParam) .* log((1-sparsityParam)./(1-rho));
JsparesWb = JWb + beta*sum(KL);
cost = JsparesWb;
%derivative computation (backpropagation)
delta3 = -(data-a3) .* sigmoidDeriv(z3);
sterm = beta .* ((-sparsityParam)./rho + (1-sparsityParam)./(1-rho));
delta2 = (W2'*delta3 + repmat(sterm,1,m)) .* sigmoidDeriv(z2);
% W1grad should be equal to the term [(1/m) \Delta W^{(1)} + \lambda W^{(1)}]
DeltaW2 = delta3*a2';
W2grad = DeltaW2./m + lambda * W2;
DeltaW1 = delta2*data';
W1grad = DeltaW1./m + lambda * W1;
Deltab2 = sum(delta3,2);
b2grad = Deltab2./m;
Deltab1 = sum(delta2,2);
b1grad = Deltab1./m;
%-------------------------------------------------------------------
% After computing the cost and gradient, we will convert the gradients back
% to a vector format (suitable for minFunc). Specifically, we will unroll
% your gradient matrices into a vector.
grad = [W1grad(:) ; W2grad(:) ; b1grad(:) ; b2grad(:)];
end
%-------------------------------------------------------------------
% Here's an implementation of the sigmoid function, which you may find useful
% in your computation of the costs and the gradients. This inputs a (row or
% column) vector (say (z1, z2, z3)) and returns (f(z1), f(z2), f(z3)).
%its derivative is given by f'(z) = f(z)(1 - f(z))
function sigm = sigmoid(x)
sigm = 1 ./ (1 + exp(-x));
end
function sigmDeriv = sigmoidDeriv(x)
sigmDeriv = sigmoid(x).*(1-sigmoid(x));
end
computeNumericalGradient.m
function numgrad = computeNumericalGradient(J, theta)
% numgrad = computeNumericalGradient(J, theta)
% theta: a vector of parameters
% J: a function that outputs a real-number. Calling y = J(theta) will return the
% function value at theta.
% Initialize numgrad with zeros
numgrad = zeros(size(theta));
%% ---------- YOUR CODE HERE --------------------------------------
% Instructions:
% Implement numerical gradient checking, and return the result in numgrad.
% (See Section 2.3 of the lecture notes.)
% You should write code so that numgrad(i) is (the numerical approximation to) the
% partial derivative of J with respect to the i-th input argument, evaluated at theta.
% I.e., numgrad(i) should be the (approximately) the partial derivative of J with
% respect to theta(i).
%
% Hint: You will probably want to compute the elements of numgrad one at a time.
EPSILON = 1e-4;
e = eye(size(theta,1));
for i = 1:length(theta)
numgrad(i) = (J(theta+EPSILON*e(:,i))-J(theta-EPSILON*e(:,i)) )/(2*EPSILON);
end
%numgrad = (J(theta+EPSILON*e)-J(theta-EPSILON*e))/(2*EPSILON)
%% ---------------------------------------------------------------
end
Output
38.0000 38.0000
12.0000 12.0000
The above two columns you get should be very similar.
(Left-Your Numerical Gradient, Right-Analytical Gradient)
2.1452e-12
Norm of the difference between numerical and analytical gradient (should be < 1e-9)
0.6114 0.6114
0.7161 0.7161
0.5624 0.5624
0.7121 0.7121
0.4735 0.4735
1.0514 1.0514
0.7083 0.7083
0.7708 0.7708
0.6504 0.6504
0.8106 0.8106
0.6423 0.6423
0.9079 0.9079
1.0147 1.0147
0.7429 0.7429
0.3668 0.3668
0.6690 0.6690
0.9579 0.9579
0.6992 0.6992
0.9788 0.9788
0.6257 0.6257
0.9534 0.9534
0.8202 0.8202
0.6193 0.6193
0.8562 0.8562
0.5223 0.5223
0.6110 0.6110
0.7159 0.7159
0.5619 0.5619
0.7120 0.7120
0.4735 0.4735
1.0542 1.0542
0.7085 0.7085
0.7704 0.7704
0.6505 0.6505
0.8103 0.8103
0.6421 0.6421
0.9083 0.9083
1.0151 1.0151
0.7428 0.7428
0.3665 0.3665
0.6686 0.6686
0.9579 0.9579
0.6992 0.6992
0.9779 0.9779
0.6260 0.6260
0.9532 0.9532
0.8208 0.8208
0.6194 0.6194
0.8565 0.8565
0.5225 0.5225
0.6114 0.6114
0.7158 0.7158
0.5625 0.5625
0.7122 0.7122
0.4718 0.4718
1.0577 1.0577
0.7091 0.7091
0.7708 0.7708
0.6512 0.6512
0.8103 0.8103
0.6423 0.6423
0.9078 0.9078
1.0156 1.0156
0.7429 0.7429
0.3654 0.3654
0.6690 0.6690
0.9569 0.9569
0.6993 0.6993
0.9762 0.9762
0.6270 0.6270
0.9560 0.9560
0.8215 0.8215
0.6194 0.6194
0.8580 0.8580
0.5237 0.5237
0.6106 0.6106
0.7139 0.7139
0.5620 0.5620
0.7106 0.7106
0.4701 0.4701
1.0559 1.0559
0.7076 0.7076
0.7692 0.7692
0.6498 0.6498
0.8089 0.8089
0.6414 0.6414
0.9052 0.9052
1.0131 1.0131
0.7411 0.7411
0.3653 0.3653
0.6682 0.6682
0.9545 0.9545
0.6979 0.6979
0.9726 0.9726
0.6253 0.6253
0.9554 0.9554
0.8191 0.8191
0.6182 0.6182
0.8575 0.8575
0.5228 0.5228
0.6115 0.6115
0.7144 0.7144
0.5633 0.5633
0.7111 0.7111
0.4728 0.4728
1.0548 1.0548
0.7079 0.7079
0.7697 0.7697
0.6500 0.6500
0.8103 0.8103
0.6421 0.6421
0.9052 0.9052
1.0135 1.0135
0.7417 0.7417
0.3675 0.3675
0.6694 0.6694
0.9558 0.9558
0.6985 0.6985
0.9742 0.9742
0.6248 0.6248
0.9568 0.9568
0.8187 0.8187
0.6188 0.6188
0.8588 0.8588
0.5226 0.5226
0.6137 0.6137
0.7174 0.7174
0.5663 0.5663
0.7139 0.7139
0.4763 0.4763
1.0570 1.0570
0.7100 0.7100
0.7726 0.7726
0.6525 0.6525
0.8134 0.8134
0.6444 0.6444
0.9073 0.9073
1.0177 1.0177
0.7447 0.7447
0.3683 0.3683
0.6724 0.6724
0.9589 0.9589
0.7009 0.7009
0.9802 0.9802
0.6265 0.6265
0.9608 0.9608
0.8209 0.8209
0.6205 0.6205
0.8605 0.8605
0.5236 0.5236
0.6124 0.6124
0.7165 0.7165
0.5659 0.5659
0.7129 0.7129
0.4760 0.4760
1.0543 1.0543
0.7086 0.7086
0.7717 0.7717
0.6517 0.6517
0.8121 0.8121
0.6430 0.6430
0.9062 0.9062
1.0180 1.0180
0.7437 0.7437
0.3652 0.3652
0.6714 0.6714
0.9568 0.9568
0.6999 0.6999
0.9801 0.9801
0.6261 0.6261
0.9580 0.9580
0.8197 0.8197
0.6183 0.6183
0.8576 0.8576
0.5227 0.5227
0.6096 0.6096
0.7138 0.7138
0.5636 0.5636
0.7102 0.7102
0.4739 0.4739
1.0484 1.0484
0.7062 0.7062
0.7689 0.7689
0.6494 0.6494
0.8094 0.8094
0.6403 0.6403
0.9045 0.9045
1.0152 1.0152
0.7408 0.7408
0.3629 0.3629
0.6688 0.6688
0.9526 0.9526
0.6974 0.6974
0.9758 0.9758
0.6245 0.6245
0.9529 0.9529
0.8175 0.8175
0.6155 0.6155
0.8545 0.8545
0.5214 0.5214
0.6116 0.6116
0.7151 0.7151
0.5625 0.5625
0.7113 0.7113
0.4724 0.4724
1.0482 1.0482
0.7072 0.7072
0.7697 0.7697
0.6497 0.6497
0.8099 0.8099
0.6416 0.6416
0.9060 0.9060
1.0133 1.0133
0.7418 0.7418
0.3666 0.3666
0.6687 0.6687
0.9559 0.9559
0.6980 0.6980
0.9780 0.9780
0.6249 0.6249
0.9531 0.9531
0.8194 0.8194
0.6178 0.6178
0.8554 0.8554
0.5211 0.5211
0.6140 0.6140
0.7185 0.7185
0.5649 0.5649
0.7149 0.7149
0.4748 0.4748
1.0560 1.0560
0.7110 0.7110
0.7731 0.7731
0.6529 0.6529
0.8133 0.8133
0.6440 0.6440
0.9113 0.9113
1.0182 1.0182
0.7454 0.7454
0.3680 0.3680
0.6715 0.6715
0.9593 0.9593
0.7015 0.7015
0.9834 0.9834
0.6280 0.6280
0.9574 0.9574
0.8237 0.8237
0.6202 0.6202
0.8602 0.8602
0.5243 0.5243
0.6139 0.6139
0.7187 0.7187
0.5650 0.5650
0.7153 0.7153
0.4737 0.4737
1.0607 1.0607
0.7117 0.7117
0.7739 0.7739
0.6532 0.6532
0.8133 0.8133
0.6443 0.6443
0.9124 0.9124
1.0204 1.0204
0.7455 0.7455
0.3673 0.3673
0.6716 0.6716
0.9587 0.9587
0.7021 0.7021
0.9837 0.9837
0.6285 0.6285
0.9586 0.9586
0.8243 0.8243
0.6199 0.6199
0.8615 0.8615
0.5253 0.5253
0.6111 0.6111
0.7144 0.7144
0.5623 0.5623
0.7111 0.7111
0.4709 0.4709
1.0580 1.0580
0.7078 0.7078
0.7699 0.7699
0.6494 0.6494
0.8094 0.8094
0.6413 0.6413
0.9078 0.9078
1.0162 1.0162
0.7413 0.7413
0.3664 0.3664
0.6686 0.6686
0.9551 0.9551
0.6988 0.6988
0.9774 0.9774
0.6249 0.6249
0.9541 0.9541
0.8194 0.8194
0.6171 0.6171
0.8577 0.8577
0.5221 0.5221
0.6110 0.6110
0.7139 0.7139
0.5634 0.5634
0.7105 0.7105
0.4728 0.4728
1.0580 1.0580
0.7073 0.7073
0.7689 0.7689
0.6488 0.6488
0.8102 0.8102
0.6413 0.6413
0.9062 0.9062
1.0142 1.0142
0.7409 0.7409
0.3682 0.3682
0.6690 0.6690
0.9563 0.9563
0.6985 0.6985
0.9765 0.9765
0.6238 0.6238
0.9556 0.9556
0.8181 0.8181
0.6176 0.6176
0.8581 0.8581
0.5209 0.5209
0.6135 0.6135
0.7175 0.7175
0.5673 0.5673
0.7139 0.7139
0.4762 0.4762
1.0609 1.0609
0.7104 0.7104
0.7724 0.7724
0.6519 0.6519
0.8142 0.8142
0.6448 0.6448
0.9085 0.9085
1.0184 1.0184
0.7446 0.7446
0.3689 0.3689
0.6722 0.6722
0.9605 0.9605
0.7012 0.7012
0.9814 0.9814
0.6260 0.6260
0.9620 0.9620
0.8213 0.8213
0.6199 0.6199
0.8613 0.8613
0.5227 0.5227
0.6128 0.6128
0.7174 0.7174
0.5680 0.5680
0.7138 0.7138
0.4758 0.4758
1.0565 1.0565
0.7101 0.7101
0.7724 0.7724
0.6521 0.6521
0.8136 0.8136
0.6446 0.6446
0.9081 0.9081
1.0217 1.0217
0.7443 0.7443
0.3662 0.3662
0.6719 0.6719
0.9595 0.9595
0.7008 0.7008
0.9805 0.9805
0.6259 0.6259
0.9606 0.9606
0.8213 0.8213
0.6185 0.6185
0.8597 0.8597
0.5231 0.5231
0.6112 0.6112
0.7160 0.7160
0.5665 0.5665
0.7125 0.7125
0.4748 0.4748
1.0508 1.0508
0.7088 0.7088
0.7713 0.7713
0.6511 0.6511
0.8121 0.8121
0.6430 0.6430
0.9074 0.9074
1.0222 1.0222
0.7428 0.7428
0.3647 0.3647
0.6709 0.6709
0.9571 0.9571
0.6997 0.6997
0.9776 0.9776
0.6256 0.6256
0.9569 0.9569
0.8207 0.8207
0.6172 0.6172
0.8580 0.8580
0.5230 0.5230
0.6129 0.6129
0.7154 0.7154
0.5632 0.5632
0.7119 0.7119
0.4713 0.4713
1.0500 1.0500
0.7078 0.7078
0.7699 0.7699
0.6503 0.6503
0.8113 0.8113
0.6417 0.6417
0.9063 0.9063
1.0117 1.0117
0.7421 0.7421
0.3672 0.3672
0.6701 0.6701
0.9531 0.9531
0.6986 0.6986
0.9767 0.9767
0.6252 0.6252
0.9574 0.9574
0.8205 0.8205
0.6184 0.6184
0.8565 0.8565
0.5208 0.5208
0.6157 0.6157
0.7199 0.7199
0.5668 0.5668
0.7165 0.7165
0.4742 0.4742
1.0584 1.0584
0.7125 0.7125
0.7745 0.7745
0.6541 0.6541
0.8156 0.8156
0.6449 0.6449
0.9120 0.9120
1.0173 1.0173
0.7464 0.7464
0.3695 0.3695
0.6738 0.6738
0.9568 0.9568
0.7030 0.7030
0.9841 0.9841
0.6287 0.6287
0.9625 0.9625
0.8255 0.8255
0.6210 0.6210
0.8623 0.8623
0.5243 0.5243
0.6143 0.6143
0.7198 0.7198
0.5660 0.5660
0.7163 0.7163
0.4739 0.4739
1.0601 1.0601
0.7123 0.7123
0.7747 0.7747
0.6537 0.6537
0.8146 0.8146
0.6448 0.6448
0.9127 0.9127
1.0199 1.0199
0.7459 0.7459
0.3691 0.3691
0.6729 0.6729
0.9580 0.9580
0.7028 0.7028
0.9857 0.9857
0.6284 0.6284
0.9597 0.9597
0.8251 0.8251
0.6195 0.6195
0.8618 0.8618
0.5239 0.5239
0.6099 0.6099
0.7142 0.7142
0.5617 0.5617
0.7104 0.7104
0.4706 0.4706
1.0534 1.0534
0.7068 0.7068
0.7691 0.7691
0.6487 0.6487
0.8092 0.8092
0.6405 0.6405
0.9073 0.9073
1.0150 1.0150
0.7400 0.7400
0.3676 0.3676
0.6680 0.6680
0.9552 0.9552
0.6979 0.6979
0.9787 0.9787
0.6240 0.6240
0.9509 0.9509
0.8187 0.8187
0.6149 0.6149
0.8554 0.8554
0.5196 0.5196
0.6102 0.6102
0.7137 0.7137
0.5626 0.5626
0.7096 0.7096
0.4718 0.4718
1.0541 1.0541
0.7064 0.7064
0.7682 0.7682
0.6485 0.6485
0.8102 0.8102
0.6406 0.6406
0.9069 0.9069
1.0132 1.0132
0.7397 0.7397
0.3692 0.3692
0.6679 0.6679
0.9568 0.9568
0.6975 0.6975
0.9774 0.9774
0.6241 0.6241
0.9524 0.9524
0.8177 0.8177
0.6157 0.6157
0.8562 0.8562
0.5193 0.5193
0.6126 0.6126
0.7167 0.7167
0.5661 0.5661
0.7126 0.7126
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0.0174 0.0174
-0.0065 -0.0065
0.0188 0.0188
-0.0161 -0.0161
0.0201 0.0201
0.0054 0.0054
0.0028 0.0028
-0.0016 -0.0016
0.0176 0.0176
-0.0029 -0.0029
0.0136 0.0136
0.0143 0.0143
0.0043 0.0043
-0.0119 -0.0119
0.0064 0.0064
-0.0058 -0.0058
0.0131 0.0131
-0.0031 -0.0031
0.0099 0.0099
0.0267 0.0267
0.0093 0.0093
0.0105 0.0105
-0.0073 -0.0073
-0.0130 -0.0130
0.0060 0.0060
0.0180 0.0180
-0.0147 -0.0147
-0.0106 -0.0106
-0.0034 -0.0034
-0.0151 -0.0151
0.0231 0.0231
-0.0130 -0.0130
0.0177 0.0177
0.0112 0.0112
0.0012 0.0012
-0.0148 -0.0148
-0.0102 -0.0102
-0.0133 -0.0133
-0.0112 -0.0112
0.0004 0.0004
0.0173 0.0173
0.0090 0.0090
-0.0202 -0.0202
0.0007 0.0007
-0.0104 -0.0104
-0.0122 -0.0122
-0.0031 -0.0031
0.0094 0.0094
0.0177 0.0177
-0.0137 -0.0137
0.0163 0.0163
-0.0143 -0.0143
0.0257 0.0257
0.0001 0.0001
0.0064 0.0064
-0.0157 -0.0157
0.0235 0.0235
-0.0022 -0.0022
-0.0020 -0.0020
0.0070 0.0070
-0.0051 -0.0051
0.0090 0.0090
0.0120 0.0120
0.0151 0.0151
0.0151 0.0151
-0.0059 -0.0059
0.0160 0.0160
-0.0140 -0.0140
0.0173 0.0173
0.0039 0.0039
0.0019 0.0019
-0.0015 -0.0015
0.0133 0.0133
-0.0022 -0.0022
0.0106 0.0106
0.0114 0.0114
0.0037 0.0037
-0.0091 -0.0091
0.0051 0.0051
-0.0042 -0.0042
0.0101 0.0101
-0.0023 -0.0023
0.0078 0.0078
0.0206 0.0206
0.0074 0.0074
0.0078 0.0078
-0.0055 -0.0055
-0.0095 -0.0095
0.0050 0.0050
0.0138 0.0138
-0.0112 -0.0112
-0.0080 -0.0080
-0.0026 -0.0026
-0.0122 -0.0122
0.0173 0.0173
-0.0098 -0.0098
0.0138 0.0138
0.0086 0.0086
0.0009 0.0009
-0.0113 -0.0113
-0.0076 -0.0076
-0.0108 -0.0108
-0.0089 -0.0089
0.0005 0.0005
0.0134 0.0134
0.0070 0.0070
-0.0155 -0.0155
0.0006 0.0006
-0.0077 -0.0077
-0.0097 -0.0097
-0.0025 -0.0025
0.0075 0.0075
0.0137 0.0137
-0.0105 -0.0105
0.0123 0.0123
-0.0110 -0.0110
0.0196 0.0196
-0.0003 -0.0003
0.0047 0.0047
-0.0119 -0.0119
0.0179 0.0179
-0.0020 -0.0020
-0.0018 -0.0018
0.0052 0.0052
-0.0038 -0.0038
0.0064 0.0064
0.0089 0.0089
0.0117 0.0117
0.0116 0.0116
-0.0047 -0.0047
0.0120 0.0120
-0.0107 -0.0107
0.0134 0.0134
0.0052 0.0052
0.0026 0.0026
-0.0017 -0.0017
0.0187 0.0187
-0.0022 -0.0022
0.0148 0.0148
0.0152 0.0152
0.0043 0.0043
-0.0129 -0.0129
0.0065 0.0065
-0.0062 -0.0062
0.0139 0.0139
-0.0025 -0.0025
0.0111 0.0111
0.0281 0.0281
0.0096 0.0096
0.0106 0.0106
-0.0079 -0.0079
-0.0141 -0.0141
0.0058 0.0058
0.0189 0.0189
-0.0151 -0.0151
-0.0109 -0.0109
-0.0037 -0.0037
-0.0164 -0.0164
0.0237 0.0237
-0.0142 -0.0142
0.0180 0.0180
0.0115 0.0115
0.0016 0.0016
-0.0152 -0.0152
-0.0104 -0.0104
-0.0145 -0.0145
-0.0122 -0.0122
0.0002 0.0002
0.0178 0.0178
0.0094 0.0094
-0.0208 -0.0208
0.0012 0.0012
-0.0106 -0.0106
-0.0131 -0.0131
-0.0036 -0.0036
0.0098 0.0098
0.0187 0.0187
-0.0141 -0.0141
0.0176 0.0176
-0.0147 -0.0147
0.0269 0.0269
-0.0002 -0.0002
0.0062 0.0062
-0.0168 -0.0168
0.0246 0.0246
-0.0020 -0.0020
-0.0016 -0.0016
0.0077 0.0077
-0.0053 -0.0053
0.0093 0.0093
0.0121 0.0121
0.0155 0.0155
0.0158 0.0158
-0.0058 -0.0058
0.0172 0.0172
-0.0143 -0.0143
0.0182 0.0182
0.0035 0.0035
0.0018 0.0018
-0.0012 -0.0012
0.0114 0.0114
-0.0016 -0.0016
0.0092 0.0092
0.0096 0.0096
0.0028 0.0028
-0.0076 -0.0076
0.0042 0.0042
-0.0038 -0.0038
0.0086 0.0086
-0.0016 -0.0016
0.0069 0.0069
0.0174 0.0174
0.0059 0.0059
0.0069 0.0069
-0.0046 -0.0046
-0.0083 -0.0083
0.0041 0.0041
0.0119 0.0119
-0.0093 -0.0093
-0.0068 -0.0068
-0.0023 -0.0023
-0.0098 -0.0098
0.0149 0.0149
-0.0084 -0.0084
0.0116 0.0116
0.0072 0.0072
0.0006 0.0006
-0.0097 -0.0097
-0.0066 -0.0066
-0.0087 -0.0087
-0.0074 -0.0074
0.0001 0.0001
0.0111 0.0111
0.0056 0.0056
-0.0134 -0.0134
0.0001 0.0001
-0.0068 -0.0068
-0.0078 -0.0078
-0.0021 -0.0021
0.0058 0.0058
0.0113 0.0113
-0.0090 -0.0090
0.0105 0.0105
-0.0095 -0.0095
0.0165 0.0165
-0.0000 -0.0000
0.0039 0.0039
-0.0105 -0.0105
0.0149 0.0149
-0.0014 -0.0014
-0.0011 -0.0011
0.0048 0.0048
-0.0031 -0.0031
0.0055 0.0055
0.0075 0.0075
0.0095 0.0095
0.0095 0.0095
-0.0037 -0.0037
0.0107 0.0107
-0.0085 -0.0085
0.0116 0.0116
1.2258 1.2258
1.4346 1.4346
1.1304 1.1304
1.4271 1.4271
0.9456 0.9456
2.1116 2.1116
1.4193 1.4193
1.5447 1.5447
1.3035 1.3035
1.6257 1.6257
1.2868 1.2868
1.8187 1.8187
2.0350 2.0350
1.4874 1.4874
0.7345 0.7345
1.3427 1.3427
1.9160 1.9160
1.4011 1.4011
1.9598 1.9598
1.2543 1.2543
1.9177 1.9177
1.6439 1.6439
1.2393 1.2393
1.7196 1.7196
1.0469 1.0469
0.0097 0.0097
0.0051 0.0051
-0.0028 -0.0028
0.0319 0.0319
-0.0043 -0.0043
0.0252 0.0252
0.0265 0.0265
0.0079 0.0079
-0.0215 -0.0215
0.0116 0.0116
-0.0103 -0.0103
0.0237 0.0237
-0.0050 -0.0050
0.0184 0.0184
0.0481 0.0481
0.0166 0.0166
0.0185 0.0185
-0.0133 -0.0133
-0.0236 -0.0236
0.0104 0.0104
0.0320 0.0320
-0.0265 -0.0265
-0.0192 -0.0192
-0.0065 -0.0065
-0.0276 -0.0276
0.0410 0.0410
-0.0239 -0.0239
0.0313 0.0313
0.0197 0.0197
0.0021 0.0021
-0.0266 -0.0266
-0.0181 -0.0181
-0.0241 -0.0241
-0.0203 -0.0203
0.0006 0.0006
0.0308 0.0308
0.0161 0.0161
-0.0361 -0.0361
0.0016 0.0016
-0.0183 -0.0183
-0.0219 -0.0219
-0.0057 -0.0057
0.0169 0.0169
0.0320 0.0320
-0.0243 -0.0243
0.0295 0.0295
-0.0257 -0.0257
0.0460 0.0460
-0.0004 -0.0004
0.0108 0.0108
-0.0286 -0.0286
0.0422 0.0422
-0.0036 -0.0036
-0.0033 -0.0033
0.0126 0.0126
-0.0092 -0.0092
0.0157 0.0157
0.0209 0.0209
0.0268 0.0268
0.0272 0.0272
-0.0100 -0.0100
0.0290 0.0290
-0.0250 -0.0250
0.0310 0.0310
6.5168e-11
Iteration FunEvals Step Length Function Val Opt Cond
1 4 7.62103e-02 8.30124e+00 3.00783e+02
2 5 1.00000e+00 3.97362e+00 1.58224e+02
3 6 1.00000e+00 1.62999e+00 6.85017e+01
4 7 1.00000e+00 8.42707e-01 3.02282e+01
5 8 1.00000e+00 5.80811e-01 1.13141e+01
6 9 1.00000e+00 5.25522e-01 3.26425e+00
7 10 1.00000e+00 5.19643e-01 5.61751e-01
8 11 1.00000e+00 5.19454e-01 7.04955e-02
9 12 1.00000e+00 5.19444e-01 6.01546e-02
10 13 1.00000e+00 5.19422e-01 9.96780e-02
11 14 1.00000e+00 5.19405e-01 8.61858e-02
12 15 1.00000e+00 5.19382e-01 9.77521e-02
13 16 1.00000e+00 5.19331e-01 1.04796e-01
14 17 1.00000e+00 5.19212e-01 2.39656e-01
15 18 1.00000e+00 5.18870e-01 4.80287e-01
16 19 1.00000e+00 5.18236e-01 7.25454e-01
17 20 1.00000e+00 5.17372e-01 8.41066e-01
18 22 4.56970e-01 5.16859e-01 7.28002e-01
19 23 1.00000e+00 5.16322e-01 3.40077e-01
20 24 1.00000e+00 5.16195e-01 2.89520e-02
21 25 1.00000e+00 5.16194e-01 2.56236e-02
22 26 1.00000e+00 5.16192e-01 3.10105e-02
23 27 1.00000e+00 5.16183e-01 5.43523e-02
24 28 1.00000e+00 5.16162e-01 8.61156e-02
25 29 1.00000e+00 5.16106e-01 1.43578e-01
26 30 1.00000e+00 5.15987e-01 2.10266e-01
27 31 1.00000e+00 5.15775e-01 3.53412e-01
28 32 1.00000e+00 5.15447e-01 2.72639e-01
29 33 1.00000e+00 5.14953e-01 1.46765e-01
30 35 2.59825e-01 5.14923e-01 1.51057e-01
31 36 1.00000e+00 5.14880e-01 5.01821e-02
32 37 1.00000e+00 5.14874e-01 4.10861e-02
33 38 1.00000e+00 5.14868e-01 4.74026e-02
34 39 1.00000e+00 5.14862e-01 4.35437e-02
35 40 1.00000e+00 5.14853e-01 5.33445e-02
36 41 1.00000e+00 5.14822e-01 8.48753e-02
37 42 1.00000e+00 5.14738e-01 1.50515e-01
38 43 1.00000e+00 5.14471e-01 3.23942e-01
39 45 6.21238e-01 5.13694e-01 9.43970e-01
40 47 3.27229e-01 5.13275e-01 1.42744e+00
41 48 1.00000e+00 5.10740e-01 1.62654e+00
42 49 1.00000e+00 5.04462e-01 1.93043e+00
43 51 3.65533e-01 5.00916e-01 1.28847e+00
44 52 1.00000e+00 4.97466e-01 1.51162e+00
45 53 1.00000e+00 4.96643e-01 1.48806e+00
46 54 1.00000e+00 4.93290e-01 4.20822e-01
47 55 1.00000e+00 4.93156e-01 2.79865e-01
48 56 1.00000e+00 4.93041e-01 2.24542e-01
49 57 1.00000e+00 4.92860e-01 1.67226e-01
50 58 1.00000e+00 4.92822e-01 2.01532e-01
51 59 1.00000e+00 4.92767e-01 1.94017e-01
52 60 1.00000e+00 4.92275e-01 3.89124e-01
53 61 1.00000e+00 4.91745e-01 5.92188e-01
54 62 1.00000e+00 4.90035e-01 9.01583e-01
55 63 1.00000e+00 4.87931e-01 1.40152e+00
56 64 1.00000e+00 4.85336e-01 1.07619e+00
57 65 1.00000e+00 4.82411e-01 7.63719e-01
58 66 1.00000e+00 4.81316e-01 5.72974e-01
59 67 1.00000e+00 4.81000e-01 3.12968e-01
60 68 1.00000e+00 4.80919e-01 1.88151e-01
61 69 1.00000e+00 4.80882e-01 1.23977e-01
62 70 1.00000e+00 4.80845e-01 1.32994e-01
63 71 1.00000e+00 4.80697e-01 2.40125e-01
64 72 1.00000e+00 4.80414e-01 3.81186e-01
65 73 1.00000e+00 4.79757e-01 5.59460e-01
66 74 1.00000e+00 4.78488e-01 8.09116e-01
67 75 1.00000e+00 4.76419e-01 8.48529e-01
68 76 1.00000e+00 4.73612e-01 6.29929e-01
69 77 1.00000e+00 4.71721e-01 7.96539e-01
70 78 1.00000e+00 4.71071e-01 4.32397e-01
71 79 1.00000e+00 4.70899e-01 1.62480e-01
72 80 1.00000e+00 4.70856e-01 1.21442e-01
73 81 1.00000e+00 4.70824e-01 1.05886e-01
74 82 1.00000e+00 4.70788e-01 1.20901e-01
75 83 1.00000e+00 4.70704e-01 1.86576e-01
76 84 1.00000e+00 4.70484e-01 3.03126e-01
77 85 1.00000e+00 4.69998e-01 4.78820e-01
78 86 1.00000e+00 4.68971e-01 6.68039e-01
79 87 1.00000e+00 4.67537e-01 1.04421e+00
80 88 1.00000e+00 4.65601e-01 7.90550e-01
81 89 1.00000e+00 4.63899e-01 2.78312e-01
82 90 1.00000e+00 4.63611e-01 1.98940e-01
83 91 1.00000e+00 4.63543e-01 1.51840e-01
84 92 1.00000e+00 4.63514e-01 8.40611e-02
85 93 1.00000e+00 4.63496e-01 7.24648e-02
86 94 1.00000e+00 4.63483e-01 9.03412e-02
87 95 1.00000e+00 4.63438e-01 1.44363e-01
88 96 1.00000e+00 4.63330e-01 2.29618e-01
89 97 1.00000e+00 4.63062e-01 3.61509e-01
90 98 1.00000e+00 4.62500e-01 5.49332e-01
91 99 1.00000e+00 4.61601e-01 6.75908e-01
92 100 1.00000e+00 4.60720e-01 6.90258e-01
93 101 1.00000e+00 4.59944e-01 2.50629e-01
94 102 1.00000e+00 4.59809e-01 9.31309e-02
95 103 1.00000e+00 4.59797e-01 6.30603e-02
96 104 1.00000e+00 4.59786e-01 5.94369e-02
97 105 1.00000e+00 4.59771e-01 8.10432e-02
98 106 1.00000e+00 4.59738e-01 1.22058e-01
99 107 1.00000e+00 4.59668e-01 2.03348e-01
100 108 1.00000e+00 4.59528e-01 2.35815e-01
101 109 1.00000e+00 4.59133e-01 2.72904e-01
102 111 5.07081e-01 4.58751e-01 5.79162e-01
103 112 1.00000e+00 4.57947e-01 2.97789e-01
104 113 1.00000e+00 4.57332e-01 2.72797e-01
105 114 1.00000e+00 4.57202e-01 2.12209e-01
106 115 1.00000e+00 4.57118e-01 7.71211e-02
107 116 1.00000e+00 4.57109e-01 4.93795e-02
108 117 1.00000e+00 4.57102e-01 4.72018e-02
109 118 1.00000e+00 4.57089e-01 8.46050e-02
110 119 1.00000e+00 4.57063e-01 1.40786e-01
111 120 1.00000e+00 4.57000e-01 2.27287e-01
112 121 1.00000e+00 4.56860e-01 3.39462e-01
113 122 1.00000e+00 4.56571e-01 4.70972e-01
114 123 1.00000e+00 4.56177e-01 6.33154e-01
115 124 1.00000e+00 4.55638e-01 4.51326e-01
116 125 1.00000e+00 4.55177e-01 1.10737e-01
117 126 1.00000e+00 4.55134e-01 6.48216e-02
118 127 1.00000e+00 4.55127e-01 7.02911e-02
119 128 1.00000e+00 4.55121e-01 4.88463e-02
120 129 1.00000e+00 4.55108e-01 5.38970e-02
121 130 1.00000e+00 4.55089e-01 9.47840e-02
122 131 1.00000e+00 4.55040e-01 1.63317e-01
123 132 1.00000e+00 4.54933e-01 2.49930e-01
124 133 1.00000e+00 4.54697e-01 3.48737e-01
125 134 1.00000e+00 4.54307e-01 4.06098e-01
126 136 1.65606e-01 4.54236e-01 4.71231e-01
127 137 1.00000e+00 4.53883e-01 2.96552e-01
128 138 1.00000e+00 4.53718e-01 9.06103e-02
129 139 1.00000e+00 4.53700e-01 5.37756e-02
130 140 1.00000e+00 4.53695e-01 3.80832e-02
131 141 1.00000e+00 4.53693e-01 3.55991e-02
132 142 1.00000e+00 4.53678e-01 6.10583e-02
133 143 1.00000e+00 4.53649e-01 1.02889e-01
134 144 1.00000e+00 4.53579e-01 1.59057e-01
135 145 1.00000e+00 4.53433e-01 2.53711e-01
136 146 1.00000e+00 4.53175e-01 2.99363e-01
137 147 1.00000e+00 4.52869e-01 3.99262e-01
138 148 1.00000e+00 4.52563e-01 1.62342e-01
139 149 1.00000e+00 4.52497e-01 6.57487e-02
140 150 1.00000e+00 4.52489e-01 3.45642e-02
141 151 1.00000e+00 4.52487e-01 2.75229e-02
142 152 1.00000e+00 4.52479e-01 3.90108e-02
143 153 1.00000e+00 4.52458e-01 7.56998e-02
144 154 1.00000e+00 4.52406e-01 1.29102e-01
145 155 1.00000e+00 4.52289e-01 2.01754e-01
146 156 1.00000e+00 4.52056e-01 2.57430e-01
147 157 1.00000e+00 4.51956e-01 5.27313e-01
148 158 1.00000e+00 4.51653e-01 2.03362e-01
149 159 1.00000e+00 4.51560e-01 8.05170e-02
150 160 1.00000e+00 4.51539e-01 3.60675e-02
151 161 1.00000e+00 4.51537e-01 3.27526e-02
152 162 1.00000e+00 4.51533e-01 3.49041e-02
153 163 1.00000e+00 4.51516e-01 7.68651e-02
154 164 1.00000e+00 4.51494e-01 1.04982e-01
155 165 1.00000e+00 4.51429e-01 1.56469e-01
156 166 1.00000e+00 4.51314e-01 2.56216e-01
157 167 1.00000e+00 4.51131e-01 2.39171e-01
158 168 1.00000e+00 4.50894e-01 1.52287e-01
159 170 4.36662e-01 4.50820e-01 1.37447e-01
160 171 1.00000e+00 4.50767e-01 6.21185e-02
161 172 1.00000e+00 4.50761e-01 3.47862e-02
162 173 1.00000e+00 4.50758e-01 2.67225e-02
163 174 1.00000e+00 4.50753e-01 3.06906e-02
164 175 1.00000e+00 4.50742e-01 5.48427e-02
165 176 1.00000e+00 4.50726e-01 7.92165e-02
166 177 1.00000e+00 4.50689e-01 1.06632e-01
167 178 1.00000e+00 4.50549e-01 1.69769e-01
168 179 1.00000e+00 4.50478e-01 3.06533e-01
169 180 1.00000e+00 4.50261e-01 1.74560e-01
170 181 1.00000e+00 4.50123e-01 1.17526e-01
171 182 1.00000e+00 4.50065e-01 6.40440e-02
172 183 1.00000e+00 4.50064e-01 1.01525e-01
173 184 1.00000e+00 4.50051e-01 4.21317e-02
174 185 1.00000e+00 4.50049e-01 2.39221e-02
175 186 1.00000e+00 4.50046e-01 2.77753e-02
176 187 1.00000e+00 4.50043e-01 3.88788e-02
177 188 1.00000e+00 4.50035e-01 5.81868e-02
178 189 1.00000e+00 4.50017e-01 9.74323e-02
179 190 1.00000e+00 4.49979e-01 1.19830e-01
180 191 1.00000e+00 4.49870e-01 1.63561e-01
181 192 1.00000e+00 4.49766e-01 3.06751e-01
182 193 1.00000e+00 4.49592e-01 1.43208e-01
183 194 1.00000e+00 4.49506e-01 6.23250e-02
184 195 1.00000e+00 4.49493e-01 4.78384e-02
185 196 1.00000e+00 4.49491e-01 3.62732e-02
186 197 1.00000e+00 4.49489e-01 2.17806e-02
187 198 1.00000e+00 4.49488e-01 1.95002e-02
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399 415 1.00000e+00 4.47002e-01 2.56114e-02
400 416 1.00000e+00 4.46999e-01 2.82649e-02
Exceeded Maximum Number of Iterations