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rlm@2:
rlm@2: <h1 class="title">Building a Classical Mechanics Library in Clojure</h1>
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2: <div id="table-of-contents">
rlm@2: <h2>Table of Contents</h2>
rlm@2: <div id="text-table-of-contents">
rlm@2: <ul>
rlm@2: <li><a href="#sec-1">1 Useful data types </a>
rlm@2: <ul>
rlm@2: <li><a href="#sec-1-1">1.1 Complex numbers </a></li>
rlm@2: <li><a href="#sec-1-2">1.2 Power series </a></li>
rlm@2: <li><a href="#sec-1-3">1.3 Tuples and tensors </a>
rlm@2: <ul>
rlm@2: <li><a href="#sec-1-3-1">1.3.1 Tuples are “sequences with spin” </a></li>
rlm@2: <li><a href="#sec-1-3-2">1.3.2 Matrices </a></li>
rlm@2: </ul>
rlm@2: </li>
rlm@2: <li><a href="#sec-1-4">1.4 Generic Operations </a></li>
rlm@2: </ul>
rlm@2: </li>
rlm@2: <li><a href="#sec-2">2 Operators and Differentiation </a>
rlm@2: <ul>
rlm@2: <li><a href="#sec-2-1">2.1 Operators </a></li>
rlm@2: <li><a href="#sec-2-2">2.2 Differential Terms and Sequences </a></li>
rlm@2: </ul>
rlm@2: </li>
rlm@2: </ul>
rlm@2: </div>
rlm@2: </div>
rlm@2:
rlm@2: <div id="outline-container-1" class="outline-2">
rlm@2: <h2 id="sec-1"><span class="section-number-2">1</span> Useful data types </h2>
rlm@2: <div class="outline-text-2" id="text-1">
rlm@2:
rlm@2:
rlm@2:
rlm@2: </div>
rlm@2:
rlm@2: <div id="outline-container-1-1" class="outline-3">
rlm@2: <h3 id="sec-1-1"><span class="section-number-3">1.1</span> Complex numbers </h3>
rlm@2: <div class="outline-text-3" id="text-1-1">
rlm@2:
rlm@2:
rlm@2: </div>
rlm@2:
rlm@2: </div>
rlm@2:
rlm@2: <div id="outline-container-1-2" class="outline-3">
rlm@2: <h3 id="sec-1-2"><span class="section-number-3">1.2</span> Power series </h3>
rlm@2: <div class="outline-text-3" id="text-1-2">
rlm@2:
rlm@2:
rlm@2: </div>
rlm@2:
rlm@2: </div>
rlm@2:
rlm@2: <div id="outline-container-1-3" class="outline-3">
rlm@2: <h3 id="sec-1-3"><span class="section-number-3">1.3</span> Tuples and tensors </h3>
rlm@2: <div class="outline-text-3" id="text-1-3">
rlm@2:
rlm@2:
rlm@2:
rlm@2: </div>
rlm@2:
rlm@2: <div id="outline-container-1-3-1" class="outline-4">
rlm@2: <h4 id="sec-1-3-1"><span class="section-number-4">1.3.1</span> Tuples are “sequences with spin” </h4>
rlm@2: <div class="outline-text-4" id="text-1-3-1">
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2: <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)
rlm@2:
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">Let some objects have spin</span>
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defprotocol</span> <span style="color: #f0dfaf;">Spinning</span>
rlm@2: (up? [this])
rlm@2: (down? [this]))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">spin</span>
rlm@2: <span style="color: #8fb28f;">"Returns the spin of the Spinning s, either :up or :down"</span>
rlm@2: [<span style="color: #dfdfbf; font-weight: bold;">#^Spinning</span> s]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">cond</span> (up? s) <span style="color: #8cd0d3;">:up</span> (down? s) <span style="color: #8cd0d3;">:down</span>))
rlm@2:
rlm@2:
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">DEFINITION: A tuple is a sequence with spin</span>
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">Tuple</span>
rlm@2: [spin coll]
rlm@2:
rlm@2: clojure.lang.Seqable
rlm@2: (<span style="color: #8cd0d3;">seq</span> [this] (<span style="color: #8cd0d3;">seq</span> (.coll this)))
rlm@2:
rlm@2: clojure.lang.Counted
rlm@2: (<span style="color: #8cd0d3;">count</span> [this] (<span style="color: #8cd0d3;">count</span> (.coll this)))
rlm@2:
rlm@2: Spinning
rlm@2: (up? [this] (<span style="color: #8cd0d3;">=</span> <span style="color: #8cd0d3;">::up</span> (.spin this)))
rlm@2: (down? [this] (<span style="color: #8cd0d3;">=</span> <span style="color: #8cd0d3;">::down</span> (.spin this))))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">print-method</span> Tuple
rlm@2: [o w]
rlm@2: (<span style="color: #8cd0d3;">print-simple</span>
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">if</span> (up? o)
rlm@2: (<span style="color: #8cd0d3;">str</span> <span style="color: #cc9393;">"u"</span> (.coll o))
rlm@2: (<span style="color: #8cd0d3;">str</span> <span style="color: #cc9393;">"d"</span> (<span style="color: #8cd0d3;">vec</span>(.coll o))))
rlm@2: w))
rlm@2:
rlm@2:
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">CONSTRUCTORS</span>
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">up</span>
rlm@2: <span style="color: #8fb28f;">"Create a new up-tuple containing the contents of coll."</span>
rlm@2: [coll]
rlm@2: (Tuple. <span style="color: #8cd0d3;">::up</span> coll))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">down</span>
rlm@2: <span style="color: #8fb28f;">"Create a new down-tuple containing the contents of coll."</span>
rlm@2: [coll]
rlm@2: (Tuple. <span style="color: #8cd0d3;">::down</span> coll))
rlm@2: </pre>
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2: </div>
rlm@2:
rlm@2: </div>
rlm@2:
rlm@2: <div id="outline-container-1-3-2" class="outline-4">
rlm@2: <h4 id="sec-1-3-2"><span class="section-number-4">1.3.2</span> Matrices </h4>
rlm@2: <div class="outline-text-4" id="text-1-3-2">
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2: <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)
rlm@2: (<span style="color: #8cd0d3;">require</span> 'incanter.core) <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">use incanter's fast matrices</span>
rlm@2:
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">all-equal?</span> [coll]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">empty?</span> (<span style="color: #8cd0d3;">rest</span> coll)) true
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">and</span> (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">first</span> coll) (<span style="color: #8cd0d3;">second</span> coll))
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">recur</span> (<span style="color: #8cd0d3;">rest</span> coll)))))
rlm@2:
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defprotocol</span> <span style="color: #f0dfaf;">Matrix</span>
rlm@2: (rows [matrix])
rlm@2: (cols [matrix])
rlm@2: (diagonal [matrix])
rlm@2: (trace [matrix])
rlm@2: (determinant [matrix])
rlm@2: (transpose [matrix])
rlm@2: (conjugate [matrix])
rlm@2: )
rlm@2:
rlm@2: (<span style="color: #8cd0d3;">extend-protocol</span> Matrix incanter.Matrix
rlm@2: (rows [rs] (<span style="color: #8cd0d3;">map</span> down (<span style="color: #8cd0d3;">apply</span> map vector (<span style="color: #8cd0d3;">apply</span> map vector rs))))
rlm@2: (cols [rs] (<span style="color: #8cd0d3;">map</span> up (<span style="color: #8cd0d3;">apply</span> map vector rs)))
rlm@2: (diagonal [matrix] (incanter.core/diag matrix) )
rlm@2: (determinant [matrix] (incanter.core/det matrix))
rlm@2: (trace [matrix] (incanter.core/trace matrix))
rlm@2: (transpose [matrix] (incanter.core/trans matrix)))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">count-rows</span> [matrix]
rlm@2: ((<span style="color: #8cd0d3;">comp</span> count rows) matrix))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">count-cols</span> [matrix]
rlm@2: ((<span style="color: #8cd0d3;">comp</span> count cols) matrix))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">square?</span> [matrix]
rlm@2: (<span style="color: #8cd0d3;">=</span> (count-rows matrix) (count-cols matrix)))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">identity-matrix</span>
rlm@2: <span style="color: #8fb28f;">"Define a square matrix of size n-by-n with 1s along the diagonal and</span>
rlm@2: <span style="color: #8fb28f;"> 0s everywhere else."</span>
rlm@2: [n]
rlm@2: (incanter.core/identity-matrix n))
rlm@2:
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">matrix-by-rows</span>
rlm@2: <span style="color: #8fb28f;">"Define a matrix by giving its rows."</span>
rlm@2: [& rows]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">if</span>
rlm@2: (<span style="color: #8cd0d3;">not</span> (all-equal? (<span style="color: #8cd0d3;">map</span> count rows)))
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"All rows in a matrix must have the same number of elements."</span>))
rlm@2: (incanter.core/matrix (<span style="color: #8cd0d3;">vec</span> rows))))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">matrix-by-cols</span>
rlm@2: <span style="color: #8fb28f;">"Define a matrix by giving its columns"</span>
rlm@2: [& cols]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not</span> (all-equal? (<span style="color: #8cd0d3;">map</span> count cols)))
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"All columns in a matrix must have the same number of elements."</span>))
rlm@2: (incanter.core/matrix (<span style="color: #8cd0d3;">vec</span> (<span style="color: #8cd0d3;">apply</span> map vector cols)))))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">identity-matrix</span>
rlm@2: <span style="color: #8fb28f;">"Define a square matrix of size n-by-n with 1s along the diagonal and</span>
rlm@2: <span style="color: #8fb28f;"> 0s everywhere else."</span>
rlm@2: [n]
rlm@2: (incanter.core/identity-matrix n))
rlm@2:
rlm@2: </pre>
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2: </div>
rlm@2: </div>
rlm@2:
rlm@2: </div>
rlm@2:
rlm@2: <div id="outline-container-1-4" class="outline-3">
rlm@2: <h3 id="sec-1-4"><span class="section-number-3">1.4</span> Generic Operations </h3>
rlm@2: <div class="outline-text-3" id="text-1-4">
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2: <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)
rlm@2: (<span style="color: #8cd0d3;">use</span> 'clojure.contrib.generic.arithmetic
rlm@2: 'clojure.contrib.generic.collection
rlm@2: 'clojure.contrib.generic.functor
rlm@2: 'clojure.contrib.generic.math-functions)
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">numbers?</span>
rlm@2: <span style="color: #8fb28f;">"Returns true if all arguments are numbers, else false."</span>
rlm@2: [& xs]
rlm@2: (<span style="color: #8cd0d3;">every?</span> number? xs))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">tuple-surgery</span>
rlm@2: <span style="color: #8fb28f;">"Applies the function f to the items of tuple and the additional</span>
rlm@2: <span style="color: #8fb28f;"> arguments, if any. Returns a Tuple of the same type as tuple."</span>
rlm@2: [tuple f & xs]
rlm@2: ((<span style="color: #f0dfaf; font-weight: bold;">if</span> (up? tuple) up down)
rlm@2: (<span style="color: #8cd0d3;">apply</span> f (<span style="color: #8cd0d3;">seq</span> tuple) xs)))
rlm@2:
rlm@2:
rlm@2:
rlm@2: <span style="color: #708070;">;;; </span><span style="color: #7f9f7f;">CONTRACTION collapses two compatible tuples into a number.</span>
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">contractible?</span>
rlm@2: <span style="color: #8fb28f;">"Returns true if the tuples a and b are compatible for contraction,</span>
rlm@2: <span style="color: #8fb28f;"> else false. Tuples are compatible if they have the same number of</span>
rlm@2: <span style="color: #8fb28f;"> components, they have opposite spins, and their elements are</span>
rlm@2: <span style="color: #8fb28f;"> pairwise-compatible."</span>
rlm@2: [a b]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">and</span>
rlm@2: (<span style="color: #8cd0d3;">isa?</span> (<span style="color: #8cd0d3;">type</span> a) Tuple)
rlm@2: (<span style="color: #8cd0d3;">isa?</span> (<span style="color: #8cd0d3;">type</span> b) Tuple)
rlm@2: (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">count</span> a) (<span style="color: #8cd0d3;">count</span> b))
rlm@2: (<span style="color: #8cd0d3;">not=</span> (spin a) (spin b))
rlm@2:
rlm@2: (<span style="color: #8cd0d3;">not-any?</span> false?
rlm@2: (<span style="color: #8cd0d3;">map</span> #(<span style="color: #f0dfaf; font-weight: bold;">or</span>
rlm@2: (numbers? %1 %2)
rlm@2: (contractible? %1 %2))
rlm@2: a b))))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">contract</span>
rlm@2: <span style="color: #8fb28f;">"Contracts two tuples, returning the sum of the</span>
rlm@2: <span style="color: #8fb28f;"> products of the corresponding items. Contraction is recursive on</span>
rlm@2: <span style="color: #8fb28f;"> nested tuples."</span>
rlm@2: [a b]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not</span> (contractible? a b))
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">throw</span>
rlm@2: (Exception. <span style="color: #cc9393;">"Not compatible for contraction."</span>))
rlm@2: (<span style="color: #8cd0d3;">reduce</span> +
rlm@2: (<span style="color: #8cd0d3;">map</span>
rlm@2: (<span style="color: #8cd0d3;">fn</span> [x y]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">if</span> (numbers? x y)
rlm@2: (<span style="color: #8cd0d3;">*</span> x y)
rlm@2: (contract x y)))
rlm@2: a b))))
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">conj</span> Tuple
rlm@2: [tuple & xs]
rlm@2: (tuple-surgery tuple #(<span style="color: #8cd0d3;">apply</span> conj % xs)))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">fmap</span> Tuple
rlm@2: [f tuple]
rlm@2: (tuple-surgery tuple (<span style="color: #8cd0d3;">partial</span> map f)))
rlm@2:
rlm@2:
rlm@2:
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO: define Scalar, and add it to the hierarchy above Number and Complex</span>
rlm@2:
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [Tuple Tuple] <span style="color: #708070;">; </span><span style="color: #7f9f7f;">tuple*tuple</span>
rlm@2: [a b]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">if</span> (contractible? a b)
rlm@2: (contract a b)
rlm@2: (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">partial</span> * a) b)))
rlm@2:
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [java.lang.Number Tuple] <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">scalar * tuple</span>
rlm@2: [a x] (fmap (<span style="color: #8cd0d3;">partial</span> * a) x))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [Tuple java.lang.Number]
rlm@2: [x a] (<span style="color: #8cd0d3;">*</span> a x))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [java.lang.Number incanter.Matrix] <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">scalar * matrix</span>
rlm@2: [x M] (incanter.core/mult x M))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [incanter.Matrix java.lang.Number]
rlm@2: [M x] (<span style="color: #8cd0d3;">*</span> x M))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [incanter.Matrix incanter.Matrix] <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">matrix * matrix</span>
rlm@2: [M1 M2]
rlm@2: (incanter.core/mmult M1 M2))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [incanter.Matrix Tuple] <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">matrix * tuple</span>
rlm@2: [M v]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #f0dfaf; font-weight: bold;">and</span> (<span style="color: #8cd0d3;">apply</span> numbers? v) (up? v))
rlm@2: (<span style="color: #8cd0d3;">*</span> M (matrix-by-cols v))
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"Currently, you can only multiply a matrix by a tuple of *numbers*"</span>))
rlm@2: ))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [Tuple incanter.Matrix] <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">tuple * Matrix</span>
rlm@2: [v M]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #f0dfaf; font-weight: bold;">and</span> (<span style="color: #8cd0d3;">apply</span> numbers? v) (down? v))
rlm@2: (<span style="color: #8cd0d3;">*</span> (matrix-by-rows v) M)
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"Currently, you can only multiply a matrix by a tuple of *numbers*"</span>))
rlm@2: ))
rlm@2:
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">exp</span> incanter.Matrix
rlm@2: [M]
rlm@2: (incanter.core/exp M))
rlm@2:
rlm@2: </pre>
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2: </div>
rlm@2: </div>
rlm@2:
rlm@2: </div>
rlm@2:
rlm@2: <div id="outline-container-2" class="outline-2">
rlm@2: <h2 id="sec-2"><span class="section-number-2">2</span> Operators and Differentiation </h2>
rlm@2: <div class="outline-text-2" id="text-2">
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2: </div>
rlm@2:
rlm@2: <div id="outline-container-2-1" class="outline-3">
rlm@2: <h3 id="sec-2-1"><span class="section-number-3">2.1</span> Operators </h3>
rlm@2: <div class="outline-text-3" id="text-2-1">
rlm@2:
rlm@2:
rlm@2: </div>
rlm@2:
rlm@2: </div>
rlm@2:
rlm@2: <div id="outline-container-2-2" class="outline-3">
rlm@2: <h3 id="sec-2-2"><span class="section-number-3">2.2</span> Differential Terms and Sequences </h3>
rlm@2: <div class="outline-text-3" id="text-2-2">
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2: <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)
rlm@2: (<span style="color: #8cd0d3;">use</span> 'clojure.contrib.seq
rlm@2: 'clojure.contrib.generic.arithmetic
rlm@2: 'clojure.contrib.generic.collection
rlm@2: 'clojure.contrib.generic.math-functions)
rlm@2:
rlm@2: <span style="color: #708070;">;;</span><span style="color: #7f9f7f;">∂</span>
rlm@2:
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">DEFINITION : Differential Term</span>
rlm@2:
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">A quantity with infinitesimal components, e.g. x, dxdy, 4dydz. The</span>
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">coefficient of the quantity is returned by the 'coefficient' method,</span>
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">while the sequence of differential parameters is returned by the</span>
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">method 'partials'.</span>
rlm@2:
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">Instead of using (potentially ambiguous) letters to denote</span>
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">differential parameters (dx,dy,dz), we use integers. So, dxdz becomes [0 2].</span>
rlm@2:
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">The coefficient can be any arithmetic object; the</span>
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">partials must be a nonrepeating sorted sequence of nonnegative</span>
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">integers.</span>
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">DifferentialTerm</span> [coefficient partials])
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">differential-term</span>
rlm@2: <span style="color: #8fb28f;">"Make a differential term given pairs of coefficients and partials."</span>
rlm@2: [coefficient partials]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #f0dfaf; font-weight: bold;">and</span> (<span style="color: #8cd0d3;">coll?</span> partials) (<span style="color: #8cd0d3;">every?</span> #(<span style="color: #f0dfaf; font-weight: bold;">and</span> (<span style="color: #8cd0d3;">integer?</span> %) (<span style="color: #8cd0d3;">not</span>(<span style="color: #8cd0d3;">neg?</span> %))) partials))
rlm@2: (DifferentialTerm. coefficient (<span style="color: #8cd0d3;">set</span> partials))
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (java.lang.IllegalArgumentException. <span style="color: #cc9393;">"Partials must be a collection of integers."</span>))))
rlm@2:
rlm@2:
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">DEFINITION : Differential Sequence</span>
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">A differential sequence is a sequence of differential terms, all with different partials.</span>
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">Internally, it is a map from the partials of each term to their coefficients.</span>
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">DifferentialSeq</span>
rlm@2: [terms]
rlm@2: <span style="color: #708070;">;;</span><span style="color: #7f9f7f;">clojure.lang.IPersistentMap</span>
rlm@2: clojure.lang.Associative
rlm@2: (<span style="color: #8cd0d3;">assoc</span> [this key val]
rlm@2: (DifferentialSeq.
rlm@2: (<span style="color: #8cd0d3;">cons</span> (differential-term val key) terms)))
rlm@2: (<span style="color: #8cd0d3;">cons</span> [this x]
rlm@2: (DifferentialSeq. (<span style="color: #8cd0d3;">cons</span> x terms)))
rlm@2: (containsKey [this key]
rlm@2: (<span style="color: #8cd0d3;">not</span>(<span style="color: #8cd0d3;">nil?</span> (find-first #(<span style="color: #8cd0d3;">=</span> (.partials %) key) terms))))
rlm@2: (<span style="color: #8cd0d3;">count</span> [this] (<span style="color: #8cd0d3;">count</span> (.terms this)))
rlm@2: (<span style="color: #8cd0d3;">empty</span> [this] (DifferentialSeq. []))
rlm@2: (entryAt [this key]
rlm@2: ((<span style="color: #8cd0d3;">juxt</span> #(.partials %) #(.coefficient %))
rlm@2: (find-first #(<span style="color: #8cd0d3;">=</span> (.partials %) key) terms)))
rlm@2: (<span style="color: #8cd0d3;">seq</span> [this] (<span style="color: #8cd0d3;">seq</span> (.terms this))))
rlm@2:
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">fmap</span> DifferentialSeq
rlm@2: [f dseq]
rlm@2: (DifferentialSeq. (fmap f (.terms dseq))))
rlm@2:
rlm@2:
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">BUILDING DIFFERENTIAL OBJECTS</span>
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">differential-seq</span>
rlm@2: <span style="color: #8fb28f;">"Define a differential sequence by specifying coefficient/partials</span>
rlm@2: <span style="color: #8fb28f;">pairs, which are used to create differential terms to populate the</span>
rlm@2: <span style="color: #8fb28f;">sequence."</span>
rlm@2: ([coefficient partials]
rlm@2: (DifferentialSeq. {(<span style="color: #8cd0d3;">set</span> partials) coefficient}))
rlm@2: ([coefficient partials & cps]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">odd?</span> (<span style="color: #8cd0d3;">count</span> cps))
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"differential-seq requires an even number of terms."</span>))
rlm@2: (DifferentialSeq.
rlm@2: (<span style="color: #8cd0d3;">reduce</span>
rlm@2: #(<span style="color: #8cd0d3;">assoc</span> %1 (<span style="color: #8cd0d3;">set</span> (<span style="color: #8cd0d3;">second</span> %2)) (<span style="color: #8cd0d3;">first</span> %2))
rlm@2: {(<span style="color: #8cd0d3;">set</span> partials) coefficient}
rlm@2: (<span style="color: #8cd0d3;">partition</span> 2 cps))))))
rlm@2:
rlm@2:
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">big-part</span>
rlm@2: <span style="color: #8fb28f;">"Returns the part of the differential sequence that is finite,</span>
rlm@2: <span style="color: #8fb28f;"> i.e. not infinitely small, as a differential sequence. If given a</span>
rlm@2: <span style="color: #8fb28f;"> non-differential object, returns that object."</span>
rlm@2: [dseq]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not=</span> (<span style="color: #8cd0d3;">type</span> dseq) DifferentialSeq) dseq
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">let</span> [m (.terms dseq)
rlm@2: keys (<span style="color: #8cd0d3;">sort-by</span> count (<span style="color: #8cd0d3;">keys</span> m))
rlm@2: smallest-var (<span style="color: #8cd0d3;">last</span> (<span style="color: #8cd0d3;">last</span> keys))]
rlm@2: (DifferentialSeq.
rlm@2: (<span style="color: #8cd0d3;">reduce</span>
rlm@2: #(<span style="color: #8cd0d3;">assoc</span> %1 (<span style="color: #8cd0d3;">first</span> %2) (<span style="color: #8cd0d3;">second</span> %2))
rlm@2: {}
rlm@2: (<span style="color: #8cd0d3;">remove</span> #((<span style="color: #8cd0d3;">first</span> %) smallest-var) m))))))
rlm@2:
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">small-part</span>
rlm@2: <span style="color: #8fb28f;">"Returns the part of the differential sequence that infinitely</span>
rlm@2: <span style="color: #8fb28f;"> small, as a differential sequence. If given a non-differential</span>
rlm@2: <span style="color: #8fb28f;"> object, returns that object."</span>
rlm@2: [dseq]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not=</span> (<span style="color: #8cd0d3;">type</span> dseq) DifferentialSeq) 0
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">let</span> [m (.terms dseq)
rlm@2: keys (<span style="color: #8cd0d3;">sort-by</span> count (<span style="color: #8cd0d3;">keys</span> m))
rlm@2: smallest-var (<span style="color: #8cd0d3;">last</span> (<span style="color: #8cd0d3;">last</span> keys))]
rlm@2: (DifferentialSeq.
rlm@2: (<span style="color: #8cd0d3;">reduce</span>
rlm@2: #(<span style="color: #8cd0d3;">assoc</span> %1 (<span style="color: #8cd0d3;">first</span> %2) (<span style="color: #8cd0d3;">second</span> %2)) {}
rlm@2: (<span style="color: #8cd0d3;">filter</span> #((<span style="color: #8cd0d3;">first</span> %) smallest-var) m))))))
rlm@2:
rlm@2:
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">cartesian-product</span> [set1 set2]
rlm@2: (<span style="color: #8cd0d3;">reduce</span> concat
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">for</span> [x set1]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">for</span> [y set2]
rlm@2: [x y]))))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">differential-multiply</span>
rlm@2: <span style="color: #8fb28f;">"Multiply two differential sequences. The square of any differential</span>
rlm@2: <span style="color: #8fb28f;"> variable is zero since differential variables are infinitesimally</span>
rlm@2: <span style="color: #8fb28f;"> small."</span>
rlm@2: [dseq1 dseq2]
rlm@2: (DifferentialSeq.
rlm@2: (<span style="color: #8cd0d3;">reduce</span>
rlm@2: (<span style="color: #8cd0d3;">fn</span> [m [[vars1 coeff1] [vars2 coeff2]]]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not</span> (<span style="color: #8cd0d3;">empty?</span> (clojure.set/<span style="color: #dfdfbf; font-weight: bold;">intersection</span> vars1 vars2)))
rlm@2: m
rlm@2: (<span style="color: #8cd0d3;">assoc</span> m (clojure.set/<span style="color: #dfdfbf; font-weight: bold;">union</span> vars1 vars2) (<span style="color: #8cd0d3;">*</span> coeff1 coeff2))))
rlm@2: {}
rlm@2: (cartesian-product (.terms dseq1) (.terms dseq2)))))
rlm@2:
rlm@2:
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [DifferentialSeq DifferentialSeq]
rlm@2: [dseq1 dseq2]
rlm@2: (differential-multiply dseq1 dseq2))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">+</span> [DifferentialSeq DifferentialSeq]
rlm@2: [dseq1 dseq2]
rlm@2: (DifferentialSeq.
rlm@2: (<span style="color: #8cd0d3;">merge-with</span> + (.terms dseq1) (.terms dseq2))))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [java.lang.Number DifferentialSeq]
rlm@2: [x dseq]
rlm@2: (fmap (<span style="color: #8cd0d3;">partial</span> * x) dseq))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [DifferentialSeq java.lang.Number]
rlm@2: [dseq x]
rlm@2: (fmap (<span style="color: #8cd0d3;">partial</span> * x) dseq))
rlm@2: <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">DERIVATIVES</span>
rlm@2:
rlm@2:
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">linear-approximator</span>
rlm@2: <span style="color: #8fb28f;">"Returns an operator that linearly approximates the given function."</span>
rlm@2: ([f df|dx]
rlm@2: (<span style="color: #8cd0d3;">fn</span> [x]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">let</span> [big-part (big-part x)
rlm@2: small-part (small-part x)]
rlm@2: (<span style="color: #8cd0d3;">+</span> (fmap f big-part)
rlm@2: (<span style="color: #8cd0d3;">*</span> (fmap df|dx big-part) small-part)))))
rlm@2: ([f df|dx df|dy]
rlm@2: (<span style="color: #8cd0d3;">fn</span> [x y]
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">let</span> [X (big-part x)
rlm@2: Y (big-part y)
rlm@2: DX (small-part x)
rlm@2: DY (small-part y)]
rlm@2: (<span style="color: #8cd0d3;">+</span> (f X Y)
rlm@2: (<span style="color: #8cd0d3;">+</span> (<span style="color: #8cd0d3;">*</span> DX (df|dx X Y))
rlm@2: (<span style="color: #8cd0d3;">*</span> DY (df|dy X Y)))))))
rlm@2: )
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">D</span>[f]
rlm@2: (<span style="color: #8cd0d3;">fn</span>[x]
rlm@2: (f (differential-seq x [] 1 [0]))))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">d</span>[f partials]
rlm@2: (<span style="color: #8cd0d3;">fn</span> [x]
rlm@2: (<span style="color: #8cd0d3;">get</span>
rlm@2: (.terms ((D f)x))
rlm@2: (<span style="color: #8cd0d3;">set</span> partials)
rlm@2: 0
rlm@2: )))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">exp</span> DifferentialSeq [x]
rlm@2: ((linear-approximator exp exp) x))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">sin</span> DifferentialSeq
rlm@2: [x]
rlm@2: ((linear-approximator sin cos) x))
rlm@2:
rlm@2: (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">cos</span> DifferentialSeq
rlm@2: [x]
rlm@2: ((linear-approximator cos #(<span style="color: #8cd0d3;">-</span> (sin %))) x))
rlm@2:
rlm@2:
rlm@2:
rlm@2: </pre>
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2:
rlm@2: </div>
rlm@2: </div>
rlm@2: </div>
rlm@2: <div id="postamble">
rlm@2: <p class="date">Date: 2011-08-11 04:10:36 EDT</p>
rlm@2: <p class="author">Author: Dylan Holmes</p>
rlm@2: <p class="creator">Org version 7.6 with Emacs version 23</p>
rlm@2: <a href="http://validator.w3.org/check?uri=referer">Validate XHTML 1.0</a>
rlm@2: </div>
rlm@2: </div>
rlm@2: </body>
rlm@2: </html>