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linear-algebra.html
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<h1 id="linear-algebra-top" class="title">Linear Algebra</h1>
<p>Smile Shell provides an MATLAB like environment.
In the simplest case, you can use it as a calculator.</p>
<ul class="nav nav-tabs">
<li class="active"><a href="#java_1" data-toggle="tab">Java</a></li>
<li><a href="#scala_1" data-toggle="tab">Scala</a></li>
</ul>
<div class="tab-content">
<div class="tab-pane" id="scala_1">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
smile> "Hello, World"
res0: String = Hello, World
smile> 2
res1: Int = 2
smile> 2+3
res2: Int = 5
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_1">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
smile> "Hello, World"
$9 ==> "Hello, World"
smile> 2
$10 ==> 2
smile> 2+3
$11 ==> 5
</code></pre>
</div>
</div>
</div>
<h2 id="functions" class="title">Math Functions</h2>
<p>Besides <code>java.lang.Math</code> functions, <code>smile.math.MathEx</code>
provides many other important mathematical functions such as
<code>factorial</code>, <code>choose</code>, etc.</p>
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<li class="active"><a href="#java_2" data-toggle="tab">Java</a></li>
<li><a href="#scala_2" data-toggle="tab">Scala</a></li>
</ul>
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<div class="tab-pane" id="scala_2">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
smile> choose(10, 3)
res8: Double = 120.0
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_2">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
smile> import static smile.math.MathEx.*
smile> choose(10, 3)
$14 ==> 120.0
</code></pre>
</div>
</div>
</div>
<h2 id="special" class="title">Special Functions</h2>
<p>Special mathematical functions include <code>beta</code>,
<code>erf</code>, <code>gamma</code> and their related functions. Special
functions are particular mathematical functions which have more or less
established names and notations due to their importance in mathematical
analysis, functional analysis, physics, or other applications.
Many special functions appear as solutions of differential equations or
integrals of elementary functions. For example, the error function
<code>erf</code> (also called the Gauss error function) is a special
function of sigmoid shape which occurs in probability, statistics, materials
science, and partial differential equations. The complementary error function,
denoted <code>erfc</code>, is defined as <code>erfc(x) = 1 - erf(x)</code>.
The error function and complementary error function are special cases of the
incomplete gamma function.</p>
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<li class="active"><a href="#java_3" data-toggle="tab">Java</a></li>
<li><a href="#scala_3" data-toggle="tab">Scala</a></li>
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<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
smile> erf(1.0)
res0: Double = 0.8427007929497149
smile> digamma(1.0)
res11: Double = -0.5772156649015328
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_3">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
smile> import smile.math.special.*
smile> Erf.erf(1.0)
$16 ==> 0.8427007929497149
smile> Gamma.digamma(1.0)
$17 ==> -0.5772156649015328
</code></pre>
</div>
</div>
</div>
<h2 id="vector" class="title">Vector Operations</h2>
<p>Common arithmetic operations on vectors and scalars are similar as in R and Matlab.</p>
<ul class="nav nav-tabs">
<li class="active"><a href="#java_4" data-toggle="tab">Java</a></li>
<li><a href="#scala_4" data-toggle="tab">Scala</a></li>
</ul>
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<div class="tab-pane" id="scala_4">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
smile> val x = c(1.0, 2.0, 3.0, 4.0)
smile> val y = c(4.0, 3.0, 2.0, 1.0)
smile> x + y
res22: smile.math.VectorAddVector = Array(5.0, 5.0, 5.0, 5.0)
smile> 1.5 * x - 3.0 * y
res24: smile.math.VectorSubVector = Array(-10.5, -6.0, -1.5, 3.0)
</code></pre>
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</div>
<div class="tab-pane active" id="java_4">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
smile> double[] x = {1.0, 2.0, 3.0, 4.0}
x ==> double[4] { 1.0, 2.0, 3.0, 4.0 }
smile> double[] y = {4.0, 3.0, 2.0, 1.0}
y ==> double[4] { 4.0, 3.0, 2.0, 1.0 }
// vector expression is not supported in Java
</code></pre>
</div>
</div>
</div>
<p>Note that these operations are lazy. The computation is only performed when
the results are needed, e.g. when the expression is used where a vector is expected.
In the Shell, the expression is immediately performed because the Shell
always prints out the results.</p>
<p>For a vector, there are multiple functions to calculate its norm such as <code>norm</code> (L2 norm), <code>norm1</code> (L1 norm),
<code>norm2</code> (L2 norm), <code>normInf</code> (infinity norm), <code>normFro</code> (Frobenius norm).
We can also <code>standardize</code> a vector to mean 0 and variance 1,
<code>unitize</code> it so that L2 norm be 1,
or <code>unitize1</code> it so that L1 norm be 1.</p>
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<li class="active"><a href="#java_5" data-toggle="tab">Java</a></li>
<li><a href="#scala_5" data-toggle="tab">Scala</a></li>
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<div class="tab-pane" id="scala_5">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
smile> norm(x)
res13: Double = 5.477225575051661
smile> unitize(y)
smile> y
res14: Array[Double] = Array(0.7302967433402214, 0.5477225575051661, 0.3651483716701107, 0.18257418583505536)
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_5">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
smile> norm(x)
$20 ==> 5.477225575051661
smile> unitize(y)
smile> y
y ==> double[4] { 0.7302967433402214, 0.5477225575051661, 0.3651483716701107, 0.18257418583505536 }
</code></pre>
</div>
</div>
</div>
<p>For a pair of vectors, we can calculate the dot product, distance, divergence, covariance,
and correlations with <code>dot</code>, <code>distance</code>, <code>kld</code> (Kullback-Leibler Divergence),
<code>jsd</code> (Jensen-Shannon Divergence), <code>cov</code>, <code>cor</code> (Pearson Correlation),
<code>spearman</code> (Spearman Rank Correlation Coefficient), <code>kendall</code> (Kendall Tau Rank Correlation Coefficient).</p>
<ul class="nav nav-tabs">
<li class="active"><a href="#java_6" data-toggle="tab">Java</a></li>
<li><a href="#scala_6" data-toggle="tab">Scala</a></li>
</ul>
<div class="tab-content">
<div class="tab-pane" id="scala_6">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
smile> dot(x, y)
res16: Double = 3.651483716701107
smile> cov(x, y)
res17: Double = -0.30429030972509225
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_6">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
smile> dot(x, y)
res5: Double = 3.651483716701107
smile> cov(x, y)
res6: Double = -0.30429030972509225
</code></pre>
</div>
</div>
</div>
<h2 id="matrix" class="title">Matrix Operations</h2>
<p>Like Matlab, we can use <code>eye</code>, <code>zeros</code> and <code>ones</code>
to create identity, zero, or all-ones matrix, respectively.
To create a matrix from 2-dimensional array, we can use the constructor <code>matrix</code>
or the <code>~</code> operator.
The <code>~</code> operator can be applied to 1-dimensional array too, which creates
a single column matrix.</p>
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<div class="tab-pane" id="scala_7">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
val a = matrix(
c(0.7220180, 0.07121225, 0.6881997),
c(-0.2648886, -0.89044952, 0.3700456),
c(-0.6391588, 0.44947578, 0.6240573)
)
val b = matrix(
c(0.6881997, -0.07121225, 0.7220180),
c(0.3700456, 0.89044952, -0.2648886),
c(0.6240573, -0.44947578, -0.6391588)
)
val C = Array(
Array(0.9527204, -0.2973347, 0.06257778),
Array(-0.2808735, -0.9403636, -0.19190231),
Array(0.1159052, 0.1652528, -0.97941688)
)
val c = ~C // or val c = matrix(C)
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_7">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
import smile.math.matrix.*;
double[][] A = {
{0.7220180, 0.07121225, 0.6881997},
{-0.2648886, -0.89044952, 0.3700456},
{-0.6391588, 0.44947578, 0.6240573}
};
double[][] B = {
{0.6881997, -0.07121225, 0.7220180},
{0.3700456, 0.89044952, -0.2648886},
{0.6240573, -0.44947578, -0.6391588}
};
double[][] C = {
{0.9527204, -0.2973347, 0.06257778},
{-0.2808735, -0.9403636, -0.19190231},
{0.1159052, 0.1652528, -0.97941688}
};
var a = Matrix.of(A);
var b = Matrix.of(B);
var c = Matrix.of(C);
</code></pre>
</div>
</div>
</div>
<p>In Scala, matrix-vector operations are just like in math formula.</p>
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<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
smile> val x = c(1.0, 2.0, 3.0)
x: Array[Double] = Array(1.0, 2.0, 3.0)
smile> val y = c(3.0, 2.0, 1.0)
y: Array[Double] = Array(3.0, 2.0, 1.0)
smile> val res: Array[Double] = a * x + 1.5 * y
res: Array[Double] = Array(7.4290416, 2.06434916, 3.63196466)
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_8">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
smile> double[] x = {1.0, 2.0, 3.0}
x ==> double[3] { 1.0, 2.0, 3.0 }
smile> double[] y = {3.0, 2.0, 1.0}
y ==> double[3] { 3.0, 2.0, 1.0 }
smile> a.mv(1.0, x, 1.5, y)
$48 ==> double[3] { 7.4290416, 2.06434916, 3.63196466 }
</code></pre>
</div>
</div>
</div>
<p>Similarly, for matrix-matrix operations:</p>
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<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
smile> a + b // returns a lazy expression of a + b
res27: smile.math.MatrixAddMatrix =
1.4102 0.0000 1.4102
0.1052 0.0000 0.1052
-0.0151 0.0000 -0.0151
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_9">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
smile> a.add(b) // a = a + b
$44 ==> 3 x 3
1.4102 0.0000 1.4102
0.1052 0.0000 0.1052
-0.0151 0.0000 -0.0151
</code></pre>
</div>
</div>
</div>
<p>Note that <code>a * b</code> are element-wise:</p>
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<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
smile> a * b // returns a lazy expression of a * b
res28: smile.math.MatrixMulMatrix =
0.4969 -0.0051 0.4969
-0.0980 -0.7929 -0.0980
-0.3989 -0.2020 -0.3989
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_10">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
smile> a.mul(b) // a *= b
$45 ==> 3 x 3
0.4969 -0.0051 0.4969
-0.0980 -0.7929 -0.0980
-0.3989 -0.2020 -0.3989
</code></pre>
</div>
</div>
</div>
<p>For matrix multiplication, the operator is <code>%*%</code>, same as in R</p>
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<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
smile> a %*% b %*% c
[main] INFO smile.math.MatrixOrderOptimization - The minimum cost of matrix multiplication chain: 54
res29: smile.math.MatrixExpression =
0.9984 0.0067 0.0554
-0.0257 0.9361 0.3508
-0.0495 -0.3517 0.9348
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_11">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
smile> a.mm(b).mm(c)
$49 ==> 3 x 3
0.9984 0.0067 0.0554
-0.0257 0.9361 0.3508
-0.0495 -0.3517 0.9348
</code></pre>
</div>
</div>
</div>
<p>The method <code>Matrix.transpose</code> returns the transpose of matrix,
which executes immediately. However, the method <code>t</code> is preferred
on <code>MatrixExpression</code> as it is lazy.</p>
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<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
smile> a %*% b.t %*% c
[main] INFO smile.math.MatrixOrderOptimization - The minimum cost of matrix multiplication chain: 54
res30: smile.math.MatrixExpression =
0.8978 -0.4369 0.0543
0.4189 0.8856 0.2006
-0.1357 -0.1574 0.9782
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_12">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
smile> a.mt(b).mm(c)
$50 ==> 3 x 3
0.8978 -0.4369 0.0543
0.4189 0.8856 0.2006
-0.1357 -0.1574 0.9782
</code></pre>
</div>
</div>
</div>
<p>Smile has runtime optimization for matrix multiplication chain, which can greatly
improve the performance. Note that this optimization is only available in Scala API.
In the below we generate several random matrices and multiply them together.</p>
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<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
val a = randn( 300, 900)
val b = randn( 900, 150)
val c = randn( 150, 1800)
val d = randn(1800, 30)
time("matrix multiplication") {(a %*% b %*% c %*% d).toMatrix}
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_13">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
smile> var a = Matrix.randn( 300, 900)
[main] INFO smile.math.MathEx - Set RNG seed 19650218 for thread main
a ==> 300 x 900
-0.9299 -0.4984 1.3793 1.8589 ... 4842 -0.5907 ...
...
smile> var b = Matrix.randn( 900, 150)
b ==> 900 x 150
0.9851 0.9842 0.7543 -0.6598 ... 9706 0.9420 ...
...
smile> var c = Matrix.randn( 150, 1800)
c ==> 150 x 1800
0.8682 -1.9094 -0.2466 0.1238 ... 2070 -1.1657 ...
...
smile> var d = Matrix.randn(1800, 30)
d ==> 1800 x 30
-0.1421 -0.4016 -1.7960 0.2153 ... 6566 -1.0292 ...
...
smile> a.mm(b).mm(c).mm(d)
$55 ==> 300 x 30
1027.7940 -7083.7899 20850.3728 14316.0928 3122.5039 6656.6392 -14332.0066 ...
-15355.3544 18424.0367 3362.8806 1969.2299 -23705.3085 -8948.9324 7468.9138 ...
-442.4282 7575.2694 -8070.4564 15107.1986 10726.3271 -170.4820 -19199.5856 ...
4155.9123 -11273.9462 4326.8992 -276.7401 22746.9657 23260.6079 -1052.8137 ...
27450.9909 -353.9005 26619.2334 -2807.0904 -18675.1774 -7891.4804 9164.3414 ...
11257.9267 -12587.2370 -15836.0616 -8085.9522 -1277.4189 -11561.2331 -8508.3348 ...
-7136.4159 3785.3912 -15033.8276 9799.7746 -16499.4337 16218.9645 13444.4842 ...
...
</code></pre>
</div>
</div>
</div>
<p>where <code>randn()</code> creates a matrix of normally distributed
random numbers. The shell will try to load machine optimized
BLAS/LAPACK native libraries for most matrix computation.
If BLAS/LAPACK is not available, smile will fall back to pure Java
implementation.</p>
<h2 id="decomposition" class="title">Matrix Decomposition</h2>
<p>In linear algebra, a matrix decomposition or matrix factorization
is a factorization of a matrix into a product of matrices.
There are various matrix decompositions. In Smile, we provide
LU, QR, Cholesky, eigen, and SVD decomposition by functions
<code>lu</code>, <code>qr</code>, <code>cholesky</code>,
<code>eigen</code>, and <code>svd</code>, respectively.</p>
<p>With these decompositions, many important linear algebra operations
can be performed such as calculating matrix rank, determinant, solving
linear systems, computing inverse matrix, etc.
In fact, Smile has functions <code>det</code>,
<code>rank</code>, <code>inv</code> and operator <code>\</code>
for these common computation.</p>
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<li class="active"><a href="#java_14" data-toggle="tab">Java</a></li>
<li><a href="#scala_14" data-toggle="tab">Scala</a></li>
</ul>
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<div class="tab-pane" id="scala_14">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-scala"><code>
smile> val x = Array(1.0, 2.0, 3.0)
x: Array[Double] = Array(1.0, 2.0, 3.0)
smile> a \ x
res14: Array[Double] = Array(2.9290414582113184, -0.9356509345036078, 2.131964578605774)
smile> inv(a)
res19: smile.math.matrix.Matrix =
0.7220 -0.2649 -0.6392
0.0712 -0.8904 0.4495
0.6882 0.3700 0.6241
smile> inv(a) %*% a
res21: smile.math.MatrixExpression =
1.0000 0.0000 0.0000
-0.0000 1.0000 0.0000
-0.0000 0.0000 1.0000
</code></pre>
</div>
</div>
<div class="tab-pane active" id="java_14">
<div class="code" style="text-align: left;">
<pre class="prettyprint lang-java"><code>
smile> double[] x = {1.0, 2.0, 3.0}
x ==> double[3] { 1.0, 2.0, 3.0 }
smile> var inv = a.inverse()
inv ==> 3 x 3
0.7220 -0.2649 -0.6392
0.0712 -0.8904 0.4495
0.6882 0.3700 0.6241
smile> inv.mm(a)
$67 ==> 3 x 3
1.0000 -0.0000 0.0000
-0.0000 1.0000 0.0000
0.0000 0.0000 1.0000
smile> var lu = a.lu()
lu ==> smile.math.matrix.Matrix$LU@5f375618
smile> lu.solve(x)
$69 ==> double[3] { -1.7252356779053744, -0.3612592362819077, 3.3004624918302046 }
</code></pre>
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