update week 39

This commit is contained in:
Morten Hjorth-Jensen
2022-09-26 06:43:06 +02:00
parent f4e55b200c
commit 878f01db99
7 changed files with 7907 additions and 1526 deletions
+161 -68
View File
@@ -38,12 +38,6 @@ doconce format html week39.do.txt --html_style=bootstrap --pygments_html_style=d
{'highest level': 2,
'sections': [('Plan for week 39', 2, None, 'plan-for-week-39'),
('Thursday September 29', 2, None, 'thursday-september-29'),
('Searching for Optimal Regularization Parameters $\\lambda$',
2,
None,
'searching-for-optimal-regularization-parameters-lambda'),
('Grid Search', 2, None, 'grid-search'),
('Randomized Grid Search', 2, None, 'randomized-grid-search'),
('Optimization, the central part of any Machine Learning '
'algortithm',
2,
@@ -159,8 +153,6 @@ doconce format html week39.do.txt --html_style=bootstrap --pygments_html_style=d
2,
None,
'using-gradient-descent-methods-limitations'),
('Challenge yourself', 2, None, 'challenge-yourself'),
('Friday September 30', 2, None, 'friday-september-30'),
('Stochastic Gradient Descent',
2,
None,
@@ -181,7 +173,80 @@ doconce format html week39.do.txt --html_style=bootstrap --pygments_html_style=d
('Code with a Number of Minibatches which varies',
2,
None,
'code-with-a-number-of-minibatches-which-varies')]}
'code-with-a-number-of-minibatches-which-varies'),
('Overview video on Stochastic Gradient Descent',
2,
None,
'overview-video-on-stochastic-gradient-descent'),
('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
('Stochastic Gradient Descent (SGD)',
2,
None,
'stochastic-gradient-descent-sgd'),
('Stochastic Gradient Descent',
2,
None,
'stochastic-gradient-descent'),
('Computation of gradients', 2, None, 'computation-of-gradients'),
('SGD example', 2, None, 'sgd-example'),
('The gradient step', 2, None, 'the-gradient-step'),
('Simple example code', 2, None, 'simple-example-code'),
('When do we stop?', 2, None, 'when-do-we-stop'),
('Slightly different approach',
2,
None,
'slightly-different-approach'),
('Program for stochastic gradient',
2,
None,
'program-for-stochastic-gradient'),
('Replace or not', 2, None, 'replace-or-not'),
('Momentum based GD', 2, None, 'momentum-based-gd'),
('More on momentum based approaches',
2,
None,
'more-on-momentum-based-approaches'),
('Momentum parameter', 2, None, 'momentum-parameter'),
('Second moment of the gradient',
2,
None,
'second-moment-of-the-gradient'),
('RMS prop', 2, None, 'rms-prop'),
('ADAM optimizer', 2, None, 'adam-optimizer'),
('Practical tips', 2, None, 'practical-tips'),
('Automatic differentiation',
2,
None,
'automatic-differentiation'),
('Using autograd', 2, None, 'using-autograd'),
('Autograd with more complicated functions',
2,
None,
'autograd-with-more-complicated-functions'),
('More complicated functions using the elements of their '
'arguments directly',
2,
None,
'more-complicated-functions-using-the-elements-of-their-arguments-directly'),
('Functions using mathematical functions from Numpy',
2,
None,
'functions-using-mathematical-functions-from-numpy'),
('More autograd', 2, None, 'more-autograd'),
('And with loops', 2, None, 'and-with-loops'),
('Using recursion', 2, None, 'using-recursion'),
('Unsupported functions', 2, None, 'unsupported-functions'),
('The syntax a.dot(b) when finding the dot product',
2,
None,
'the-syntax-a-dot-b-when-finding-the-dot-product'),
('Recommended to avoid', 2, None, 'recommended-to-avoid'),
('Using Autograd with OLS', 2, None, 'using-autograd-with-ols'),
('Including Stochastic Gradient Descent with Autograd',
2,
None,
'including-stochastic-gradient-descent-with-autograd'),
('And Logistic Regression', 2, None, 'and-logistic-regression')]}
end of tocinfo -->
<body>
@@ -218,62 +283,90 @@ MathJax.Hub.Config({
<ul class="dropdown-menu">
<!-- navigation toc: --> <li><a href="._week39-bs001.html#plan-for-week-39" style="font-size: 80%;">Plan for week 39</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs002.html#thursday-september-29" style="font-size: 80%;">Thursday September 29</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs003.html#searching-for-optimal-regularization-parameters-lambda" style="font-size: 80%;">Searching for Optimal Regularization Parameters \( \lambda \)</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs004.html#grid-search" style="font-size: 80%;">Grid Search</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs005.html#randomized-grid-search" style="font-size: 80%;">Randomized Grid Search</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs006.html#optimization-the-central-part-of-any-machine-learning-algortithm" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs007.html#revisiting-our-logistic-regression-case" style="font-size: 80%;">Revisiting our Logistic Regression case</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs008.html#the-equations-to-solve" style="font-size: 80%;">The equations to solve</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs009.html#solving-using-newton-raphson-s-method" style="font-size: 80%;">Solving using Newton-Raphson's method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs010.html#brief-reminder-on-newton-raphson-s-method" style="font-size: 80%;">Brief reminder on Newton-Raphson's method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs011.html#the-equations" style="font-size: 80%;">The equations</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs012.html#simple-geometric-interpretation" style="font-size: 80%;">Simple geometric interpretation</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs013.html#extending-to-more-than-one-variable" style="font-size: 80%;">Extending to more than one variable</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs014.html#steepest-descent" style="font-size: 80%;">Steepest descent</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs015.html#more-on-steepest-descent" style="font-size: 80%;">More on Steepest descent</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs016.html#the-ideal" style="font-size: 80%;">The ideal</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs017.html#the-sensitiveness-of-the-gradient-descent" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs018.html#convex-functions" style="font-size: 80%;">Convex functions</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs019.html#convex-function" style="font-size: 80%;">Convex function</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs020.html#conditions-on-convex-functions" style="font-size: 80%;">Conditions on convex functions</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs021.html#more-on-convex-functions" style="font-size: 80%;">More on convex functions</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs022.html#some-simple-problems" style="font-size: 80%;">Some simple problems</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs023.html#standard-steepest-descent" style="font-size: 80%;">Standard steepest descent</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs024.html#gradient-method" style="font-size: 80%;">Gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs026.html#steepest-descent-method" style="font-size: 80%;">Steepest descent method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs026.html#steepest-descent-method" style="font-size: 80%;">Steepest descent method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs027.html#final-expressions" style="font-size: 80%;">Final expressions</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs028.html#steepest-descent-example" style="font-size: 80%;">Steepest descent example</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs036.html#conjugate-gradient-method" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs036.html#conjugate-gradient-method" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs036.html#conjugate-gradient-method" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs036.html#conjugate-gradient-method" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs033.html#conjugate-gradient-method-and-iterations" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs036.html#conjugate-gradient-method" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs036.html#conjugate-gradient-method" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs036.html#conjugate-gradient-method" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs037.html#revisiting-our-first-homework" style="font-size: 80%;">Revisiting our first homework</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs042.html#gradient-descent-example" style="font-size: 80%;">Gradient descent example</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs039.html#the-derivative-of-the-cost-loss-function" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs040.html#the-hessian-matrix" style="font-size: 80%;">The Hessian matrix</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs041.html#simple-program" style="font-size: 80%;">Simple program</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs042.html#gradient-descent-example" style="font-size: 80%;">Gradient Descent Example</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs043.html#and-a-corresponding-example-using-scikit-learn" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
<!-- navigation toc: --> <li><a href="._week39-bs044.html#gradient-descent-and-ridge" style="font-size: 80%;">Gradient descent and Ridge</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs045.html#the-hessian-matrix-for-ridge-regression" style="font-size: 80%;">The Hessian matrix for Ridge Regression</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs046.html#program-example-for-gradient-descent-with-ridge-regression" style="font-size: 80%;">Program example for gradient descent with Ridge Regression</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs047.html#using-gradient-descent-methods-limitations" style="font-size: 80%;">Using gradient descent methods, limitations</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs048.html#challenge-yourself" style="font-size: 80%;">Challenge yourself</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs049.html#friday-september-30" style="font-size: 80%;">Friday September 30</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs050.html#stochastic-gradient-descent" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs051.html#computation-of-gradients" style="font-size: 80%;">Computation of gradients</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs052.html#sgd-example" style="font-size: 80%;">SGD example</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs053.html#the-gradient-step" style="font-size: 80%;">The gradient step</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs054.html#simple-example-code" style="font-size: 80%;">Simple example code</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs055.html#when-do-we-stop" style="font-size: 80%;">When do we stop?</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs056.html#slightly-different-approach" style="font-size: 80%;">Slightly different approach</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs057.html#program-for-stochastic-gradient" style="font-size: 80%;">Program for stochastic gradient</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs058.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;">Code with a Number of Minibatches which varies</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs003.html#optimization-the-central-part-of-any-machine-learning-algortithm" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs004.html#revisiting-our-logistic-regression-case" style="font-size: 80%;">Revisiting our Logistic Regression case</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs005.html#the-equations-to-solve" style="font-size: 80%;">The equations to solve</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs006.html#solving-using-newton-raphson-s-method" style="font-size: 80%;">Solving using Newton-Raphson's method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs007.html#brief-reminder-on-newton-raphson-s-method" style="font-size: 80%;">Brief reminder on Newton-Raphson's method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs008.html#the-equations" style="font-size: 80%;">The equations</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs009.html#simple-geometric-interpretation" style="font-size: 80%;">Simple geometric interpretation</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs010.html#extending-to-more-than-one-variable" style="font-size: 80%;">Extending to more than one variable</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs011.html#steepest-descent" style="font-size: 80%;">Steepest descent</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs012.html#more-on-steepest-descent" style="font-size: 80%;">More on Steepest descent</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs013.html#the-ideal" style="font-size: 80%;">The ideal</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs014.html#the-sensitiveness-of-the-gradient-descent" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs015.html#convex-functions" style="font-size: 80%;">Convex functions</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs016.html#convex-function" style="font-size: 80%;">Convex function</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs017.html#conditions-on-convex-functions" style="font-size: 80%;">Conditions on convex functions</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs018.html#more-on-convex-functions" style="font-size: 80%;">More on convex functions</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs019.html#some-simple-problems" style="font-size: 80%;">Some simple problems</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs020.html#standard-steepest-descent" style="font-size: 80%;">Standard steepest descent</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs021.html#gradient-method" style="font-size: 80%;">Gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs023.html#steepest-descent-method" style="font-size: 80%;">Steepest descent method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs023.html#steepest-descent-method" style="font-size: 80%;">Steepest descent method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs024.html#final-expressions" style="font-size: 80%;">Final expressions</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs025.html#steepest-descent-example" style="font-size: 80%;">Steepest descent example</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs033.html#conjugate-gradient-method" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs033.html#conjugate-gradient-method" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs033.html#conjugate-gradient-method" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs033.html#conjugate-gradient-method" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs030.html#conjugate-gradient-method-and-iterations" style="font-size: 80%;">Conjugate gradient method and iterations</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs033.html#conjugate-gradient-method" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs033.html#conjugate-gradient-method" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs033.html#conjugate-gradient-method" style="font-size: 80%;">Conjugate gradient method</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs034.html#revisiting-our-first-homework" style="font-size: 80%;">Revisiting our first homework</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs039.html#gradient-descent-example" style="font-size: 80%;">Gradient descent example</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs036.html#the-derivative-of-the-cost-loss-function" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs037.html#the-hessian-matrix" style="font-size: 80%;">The Hessian matrix</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs038.html#simple-program" style="font-size: 80%;">Simple program</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs039.html#gradient-descent-example" style="font-size: 80%;">Gradient Descent Example</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs040.html#and-a-corresponding-example-using-scikit-learn" style="font-size: 80%;">And a corresponding example using <b>scikit-learn</b></a></li>
<!-- navigation toc: --> <li><a href="._week39-bs041.html#gradient-descent-and-ridge" style="font-size: 80%;">Gradient descent and Ridge</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs042.html#the-hessian-matrix-for-ridge-regression" style="font-size: 80%;">The Hessian matrix for Ridge Regression</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs043.html#program-example-for-gradient-descent-with-ridge-regression" style="font-size: 80%;">Program example for gradient descent with Ridge Regression</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs044.html#using-gradient-descent-methods-limitations" style="font-size: 80%;">Using gradient descent methods, limitations</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs057.html#stochastic-gradient-descent" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs058.html#computation-of-gradients" style="font-size: 80%;">Computation of gradients</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs059.html#sgd-example" style="font-size: 80%;">SGD example</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs060.html#the-gradient-step" style="font-size: 80%;">The gradient step</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs061.html#simple-example-code" style="font-size: 80%;">Simple example code</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs062.html#when-do-we-stop" style="font-size: 80%;">When do we stop?</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs063.html#slightly-different-approach" style="font-size: 80%;">Slightly different approach</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs064.html#program-for-stochastic-gradient" style="font-size: 80%;">Program for stochastic gradient</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs053.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;">Code with a Number of Minibatches which varies</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs054.html#overview-video-on-stochastic-gradient-descent" style="font-size: 80%;">Overview video on Stochastic Gradient Descent</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs055.html#batches-and-mini-batches" style="font-size: 80%;">Batches and mini-batches</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs056.html#stochastic-gradient-descent-sgd" style="font-size: 80%;">Stochastic Gradient Descent (SGD)</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs057.html#stochastic-gradient-descent" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs058.html#computation-of-gradients" style="font-size: 80%;">Computation of gradients</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs059.html#sgd-example" style="font-size: 80%;">SGD example</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs060.html#the-gradient-step" style="font-size: 80%;">The gradient step</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs061.html#simple-example-code" style="font-size: 80%;">Simple example code</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs062.html#when-do-we-stop" style="font-size: 80%;">When do we stop?</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs063.html#slightly-different-approach" style="font-size: 80%;">Slightly different approach</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs064.html#program-for-stochastic-gradient" style="font-size: 80%;">Program for stochastic gradient</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs065.html#replace-or-not" style="font-size: 80%;">Replace or not</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs066.html#momentum-based-gd" style="font-size: 80%;">Momentum based GD</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs067.html#more-on-momentum-based-approaches" style="font-size: 80%;">More on momentum based approaches</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs068.html#momentum-parameter" style="font-size: 80%;">Momentum parameter</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs069.html#second-moment-of-the-gradient" style="font-size: 80%;">Second moment of the gradient</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs070.html#rms-prop" style="font-size: 80%;">RMS prop</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs071.html#adam-optimizer" style="font-size: 80%;">ADAM optimizer</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs072.html#practical-tips" style="font-size: 80%;">Practical tips</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs073.html#automatic-differentiation" style="font-size: 80%;">Automatic differentiation</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs074.html#using-autograd" style="font-size: 80%;">Using autograd</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs075.html#autograd-with-more-complicated-functions" style="font-size: 80%;">Autograd with more complicated functions</a></li>
<!-- navigation toc: --> <li><a href="._week39-bs076.html#more-complicated-functions-using-the-elements-of-their-arguments-directly" style="font-size: 80%;">More complicated functions using the elements of their arguments directly</a></li>
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@@ -299,11 +392,11 @@ MathJax.Hub.Config({
[1] <b>Department of Physics, University of Oslo</b>
</center>
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[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b>
[2] <b>Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University</b>
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<h4>Aug 23, 2022</h4>
<h4>Sep 26, 2022</h4>
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@@ -328,7 +421,7 @@ MathJax.Hub.Config({
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