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<head>
<link rel="SHORTCUT ICON" href="favicon.ico">
<link rel="stylesheet" type="text/css" href="resources/style.css">
<style>
table { font-size:14px; }
td { vertical-align:top; }
@media print
{
table { font-size:12px; }
td.navmain { font-size:26px; }
body { margin: 5mm 5mm 5mm 5mm; }
}
</style>
<title>Numeric Javascript: Documentation</title>
</head>
<body>
<!--#include file="resources/header.html" -->
<!--
This allows regression tests to run predictably:
<pre>
> numeric.seedrandom.seedrandom('1'); Math.random = numeric.seedrandom.random; Math.random();
0.2694
</pre>
-->
<table cellspacing=5 style="border:5px solid black;">
<tr><td colspan=3 align="center" style="font-size:18px;">
<b>Reference card for the <tt>numeric</tt> module</b>
<tr valign="top"><td valign="top" width="33%">
<table>
<tr><td><b>Function</b><td><b>Description</b>
<tr><td colspan=2><hr>
<tr><td><tt>abs</tt><td>Absolute value
<tr><td><tt>acos</tt><td>Arc-cosine
<tr><td><tt>add</tt><td>Pointwise sum x+y
<tr><td><tt>addeq</tt><td>Pointwise sum x+=y
<tr><td><tt>all</tt><td>All the components of x are true
<tr><td><tt>and</tt><td>Pointwise x && y
<tr><td><tt>andeq</tt><td>Pointwise x &= y
<tr><td><tt>any</tt><td>One or more of the components of x are true
<tr><td><tt>asin</tt><td>Arc-sine
<tr><td><tt>atan</tt><td>Arc-tangeant
<tr><td><tt>atan2</tt><td>Arc-tangeant (two parameters)
<tr><td><tt>band</tt><td>Pointwise x & y
<tr><td><tt>bench</tt><td>Benchmarking routine
<tr><td><tt>bnot</tt><td>Binary negation ~x
<tr><td><tt>bor</tt><td>Binary or x|y
<tr><td><tt>bxor</tt><td>Binary xor x^y
<tr><td><tt>cLU</tt><td>Coordinate matrix LU decomposition
<tr><td><tt>cLUsolve</tt><td>Coordinate matrix LU solve
<tr><td><tt>cdelsq</tt><td>Coordinate matrix Laplacian
<tr><td><tt>cdotMV</tt><td>Coordinate matrix-vector product
<tr><td><tt>ceil</tt><td>Pointwise Math.ceil(x)
<tr><td><tt>cgrid</tt><td>Coordinate grid for cdelsq
<tr><td><tt>clone</tt><td>Deep copy of Array
<tr><td><tt>cos</tt><td>Pointwise Math.cos(x)
<tr><td><tt>det</tt><td>Determinant
<tr><td><tt>diag</tt><td>Create diagonal matrix
<tr><td><tt>dim</tt><td>Get Array dimensions
<tr><td><tt>div</tt><td>Pointwise x/y
<tr><td><tt>diveq</tt><td>Pointwise x/=y
<tr><td><tt>dopri</tt><td>Numerical integration of ODE using Dormand-Prince RK method. Returns an object Dopri.
<tr><td><tt>Dopri.at</tt><td>Evaluate the ODE solution at a point
<tr><td><tt>dot</tt><td>Matrix-Matrix, Matrix-Vector and Vector-Matrix product
<tr><td><tt>eig</tt><td>Eigenvalues and eigenvectors
<tr><td><tt>epsilon</tt><td>2.220446049250313e-16
<tr><td><tt>eq</tt><td>Pointwise comparison x === y
<tr><td><tt>exp</tt><td>Pointwise Math.exp(x)
<tr><td><tt>floor</tt><td>Poinwise Math.floor(x)
<tr><td><tt>geq</tt><td>Pointwise x>=y
<tr><td><tt>getBlock</tt><td>Extract a block from a matrix
<tr><td><tt>getDiag</tt><td>Get the diagonal of a matrix
<tr><td><tt>gt</tt><td>Pointwise x>y
<tr><td><tt>identity</tt><td>Identity matrix
<tr><td><tt>imageURL</tt><td>Encode a matrix as an image URL
</table>
<td valign="top" width="33%">
<table>
<tr><td><b>Function</b><td><b>Description</b>
<tr><td colspan=2><hr>
<tr><td><tt>inv</tt><td>Matrix inverse
<tr><td><tt>isFinite</tt><td>Pointwise isFinite(x)
<tr><td><tt>isNaN</tt><td>Pointwise isNaN(x)
<tr><td><tt>largeArray</tt><td>Don't prettyPrint Arrays larger than this
<tr><td><tt>leq</tt><td>Pointwise x<=y
<tr><td><tt>linspace</tt><td>Generate evenly spaced values
<tr><td><tt>log</tt><td>Pointwise Math.log(x)
<tr><td><tt>lshift</tt><td>Pointwise x<<y
<tr><td><tt>lshifteq</tt><td>Pointwise x<<=y
<tr><td><tt>lt</tt><td>Pointwise x<y
<tr><td><tt>mapreduce</tt><td>Make a pointwise map-reduce function
<tr><td><tt>mod</tt><td>Pointwise x%y
<tr><td><tt>modeq</tt><td>Pointwise x%=y
<tr><td><tt>mul</tt><td>Pointwise x*y
<tr><td><tt>neg</tt><td>Pointwise -x
<tr><td><tt>neq</tt><td>Pointwise x!==y
<tr><td><tt>norm2</tt><td>Square root of the sum of the square of the entries of x
<tr><td><tt>norm2Squared</tt><td>Sum of squares of entries of x
<tr><td><tt>norminf</tt><td>Largest modulus entry of x
<tr><td><tt>not</tt><td>Pointwise logical negation !x
<tr><td><tt>or</tt><td>Pointwise logical or x||y
<tr><td><tt>oreq</tt><td>Pointwise x|=y
<tr><td><tt>parseCSV</tt><td>Parse a CSV file into an Array
<tr><td><tt>parseDate</tt><td>Pointwise parseDate(x)
<tr><td><tt>parseFloat</tt><td>Pointwise parseFloat(x)
<tr><td><tt>pointwise</tt><td>Create a pointwise function
<tr><td><tt>pow</tt><td>Pointwise Math.pow(x)
<tr><td><tt>precision</tt><td>Number of digits to prettyPrint
<tr><td><tt>prettyPrint</tt><td>Pretty-prints x
<tr><td><tt>random</tt><td>Create an Array of random numbers
<tr><td><tt>rep</tt><td>Create an Array by duplicating values
<tr><td><tt>round</tt><td>Pointwise Math.round(x)
<tr><td><tt>rrshift</tt><td>Pointwise x>>>y
<tr><td><tt>rrshifteq</tt><td>Pointwise x>>>=y
<tr><td><tt>rshift</tt><td>Pointwise x>>y
<tr><td><tt>rshifteq</tt><td>Pointwise x>>=y
<tr><td><tt>sLUP</tt><td>Sparse LUP decomposition
<tr><td><tt>sLUPsolve</tt><td>Sparse LUP solve
<tr><td><tt>same</tt><td>x and y are entrywise identical
<tr><td><tt>sclone</tt><td>Sparse clone Array
<tr><td><tt>sdiag</tt><td>Sparse diagonal matrix
<tr><td><tt>sdim</tt><td>Sparse dimension of Array
</table>
<td valign="top" width="33%">
<table>
<tr><td><b>Function</b><td><b>Description</b>
<tr><td colspan=2><hr>
<tr><td><tt>sdot</tt><td>Sparse matrix-matrix, matrix-vector, vector-matrix and vector-vector product.
<tr><td><tt>seedrandom</tt><td>The seedrandom module
<tr><td><tt>setBlock</tt><td>Set a block of a matrix
<tr><td><tt>sgather</tt><td>Sparse gather values
<tr><td><tt>sidentity</tt><td>Sparse identity
<tr><td><tt>sin</tt><td>Pointwise Math.sin(x)
<tr><td><tt>solveQP</tt><td>Solve a quadratic programming problem
<tr><td><tt>spline</tt><td>Create a Spline object
<tr><td><tt>Spline.at</tt><td>Evaluate the Spline at a point
<tr><td><tt>Spline.diff</tt><td>Differentiate the Spline
<tr><td><tt>Spline.roots</tt><td>Find all the roots of the Spline
<tr><td><tt>sqrt</tt><td>Pointwise Math.sqrt(x)
<tr><td><tt>sscatter</tt><td>Sparse scatter values
<tr><td><tt>stranspose</tt><td>Sparse transpose
<tr><td><tt>sub</tt><td>Pointwise x-y
<tr><td><tt>subeq</tt><td>Pointwise x-=y
<tr><td><tt>sum</tt><td>Sum all the entries of x
<tr><td><tt>svd</tt><td>Singular value decomposition
<tr><td><tt>t</tt><td>Create a tensor type T (may be complex-valued)
<tr><td><tt>T.<numericfun></tt><td>Supported <numericfun> are: abs, add, cos, diag, div, dot, exp, getBlock, getDiag, inv, log, mul, neg, norm2, setBlock, sin, sub, transpose
<tr><td><tt>T.conj</tt><td>Pointwise complex conjugate
<tr><td><tt>T.fft</tt><td>Fast Fourier transform
<tr><td><tt>T.get</tt><td>Read an entry
<tr><td><tt>T.getRow</tt><td>Get a row
<tr><td><tt>T.getRows</tt><td>Get a range of rows
<tr><td><tt>T.ifft</tt><td>Inverse FFT
<tr><td><tt>T.reciprocal</tt><td>Pointwise 1/z
<tr><td><tt>T.set</tt><td>Set an entry
<tr><td><tt>T.setRow</tt><td>Set a row
<tr><td><tt>T.setRows</tt><td>Set a range of rows
<tr><td><tt>T.transjugate</tt><td>The conjugate-transpose of a matrix
<tr><td><tt>tan</tt><td>Pointwise Math.tan(x)
<tr><td><tt>tensor</tt><td>Tensor product ret[i][j] = x[i]*y[j]
<tr><td><tt>toCSV</tt><td>Make a CSV file
<tr><td><tt>transpose</tt><td>Matrix transpose
<tr><td><tt>uncmin</tt><td>Unconstrained optimization
<tr><td><tt>version</tt><td>Version string for the numeric library
<tr><td><tt>xor</tt><td>Pointwise x^y
<tr><td><tt>xoreq</tt><td>Pointwise x^=y
</table></table>
<h1>Numerical analysis in Javascript</h1>
<a href="http://www.numericjs.com/">Numeric Javascript</a> is
library that provides many useful functions for numerical
calculations, particularly for linear algebra (vectors and matrices).
You can create vectors and matrices and multiply them:
<pre>
> A = [[1,2,3],[4,5,6]];
[[1,2,3],
[4,5,6]]
> x = [7,8,9]
[7,8,9]
> numeric.dot(A,x);
[50,122]
</pre>
The example shown above can be executed in the
<a href="http://www.numericjs.com/workshop.php">Javascript Workshop</a> or at any
Javascript prompt. The Workshop provides plotting capabilities:<br>
<img src="resources/workshop.png"><br>
The function <tt>workshop.plot()</tt> is essentially the <a href="http://code.google.com/p/flot/">flot</a>
plotting command.<br><br>
The <tt>numeric</tt> library provides functions that implement most of the usual Javascript
operators for vectors and matrices:
<pre>
> y = [10,1,2];
[10,1,2]
> numeric['+'](x,y)
[17,9,11]
> numeric['>'](x,y)
[false,true,true]
</pre>
These operators can also be called with plain Javascript function names:
<pre>
> numeric.add(x,y)
[17,9,11]
</pre>
You can also use these operators with three or more parameters:
<pre>
> numeric.add([1,2],[3,4],[5,6],[7,8])
[16,20]
</pre>
The function <tt>numeric.inv()</tt> can be used to compute the inverse of an invertible matrix:
<pre>
> A = [[1,2,3],[4,5,6],[7,1,9]]
[[1,2,3],
[4,5,6],
[7,1,9]]
> Ainv = numeric.inv(A);
[[-0.9286,0.3571,0.07143],
[-0.1429,0.2857,-0.1429],
[0.7381,-0.3095,0.07143]]
</pre>
The function <tt>numeric.prettyPrint()</tt> is used to print most of the examples in this documentation.
It formats objects, arrays and numbers so that they can be understood easily. All output is automatically
formatted using <tt>numeric.prettyPrint()</tt> when in the
<a href="http://www.numericjs.com/workshop.php">Workshop</a>. In order to present the information clearly and
succintly, the function <tt>numeric.prettyPrint()</tt> lays out matrices so that all the numbers align.
Furthermore, numbers are given approximately using the <tt>numeric.precision</tt> variable:
<pre>
> numeric.precision = 10; x = 3.141592653589793
3.141592654
> numeric.precision = 4; x
3.142
</pre>
The default precision is 4 digits. In addition to printing approximate numbers,
the function <tt>numeric.prettyPrint()</tt> will replace large arrays with the string <tt>...Large Array...</tt>:
<pre>
> numeric.identity(100)
...Large Array...
</pre>
By default, this happens with the Array's length is more than 50. This can be controlled by setting the
variable <tt>numeric.largeArray</tt> to an appropriate value:
<pre>
> numeric.largeArray = 2; A = numeric.identity(4)
...Large Array...
> numeric.largeArray = 50; A
[[1,0,0,0],
[0,1,0,0],
[0,0,1,0],
[0,0,0,1]]
</pre>
In particular, if you want to print all Arrays regardless of size, set <tt>numeric.largeArray = Infinity</tt>.
<br><br>
<h1>Math Object functions</h1>
The <tt>Math</tt> object functions have also been adapted to work on Arrays as follows:
<pre>
> numeric.exp([1,2]);
[2.718,7.389]
> numeric.exp([[1,2],[3,4]])
[[2.718, 7.389],
[20.09, 54.6]]
> numeric.abs([-2,3])
[2,3]
> numeric.acos([0.1,0.2])
[1.471,1.369]
> numeric.asin([0.1,0.2])
[0.1002,0.2014]
> numeric.atan([1,2])
[0.7854,1.107]
> numeric.atan2([1,2],[3,4])
[0.3218,0.4636]
> numeric.ceil([-2.2,3.3])
[-2,4]
> numeric.floor([-2.2,3.3])
[-3,3]
> numeric.log([1,2])
[0,0.6931]
> numeric.pow([2,3],[0.25,7.1])
[1.189,2441]
> numeric.round([-2.2,3.3])
[-2,3]
> numeric.sin([1,2])
[0.8415,0.9093]
> numeric.sqrt([1,2])
[1,1.414]
> numeric.tan([1,2])
[1.557,-2.185]
</pre>
<h1>Utility functions</h1>
The function <tt>numeric.dim()</tt> allows you to compute the dimensions of an Array.
<pre>
> numeric.dim([1,2])
[2]
> numeric.dim([[1,2,3],[4,5,6]])
[2,3]
</pre>
You can perform a deep comparison of Arrays using <tt>numeric.same()</tt>:
<pre>
> numeric.same([1,2],[1,2])
true
> numeric.same([1,2],[1,2,3])
false
> numeric.same([1,2],[[1],[2]])
false
> numeric.same([[1,2],[3,4]],[[1,2],[3,4]])
true
> numeric.same([[1,2],[3,4]],[[1,2],[3,5]])
false
> numeric.same([[1,2],[2,4]],[[1,2],[3,4]])
false
</pre>
You can create a multidimensional Array from a given value using <tt>numeric.rep()</tt>
<pre>
> numeric.rep([3],5)
[5,5,5]
> numeric.rep([2,3],0)
[[0,0,0],
[0,0,0]]
</pre>
You can loop over Arrays as you normally would. However, in order to quickly generate optimized
loops, the <tt>numeric</tt> library provides a few efficient loop-generation mechanisms. For example, the
<tt>numeric.mapreduce()</tt> function can be used to make a function that computes the sum of all the
entries of an Array.
<pre>
> sum = numeric.mapreduce('accum += xi','0'); sum([1,2,3])
6
> sum([[1,2,3],[4,5,6]])
21
</pre>
The functions <tt>numeric.any()</tt> and <tt>numeric.all()</tt> allow you to check whether any or all entries
of an Array are boolean true values.
<pre>
> numeric.any([false,true])
true
> numeric.any([[0,0,3.14],[0,false,0]])
true
> numeric.any([0,0,false])
false
> numeric.all([false,true])
false
> numeric.all([[1,4,3.14],["no",true,-1]])
true
> numeric.all([0,0,false])
false
</pre>
You can also create "pointwise" functions. This is how you create a vector sum function:
<pre>
> add = numeric.pointwise(['x[i]','y[i]'],'ret[i] = x[i]+y[i];'); add([1,2],[3,4])
[4,6]
</pre>
You can create a diagonal matrix using <tt>numeric.diag()</tt>
<pre>
> numeric.diag([1,2,3])
[[1,0,0],
[0,2,0],
[0,0,3]]
</pre>
The function <tt>numeric.identity()</tt> returns the identity matrix.
<pre>
> numeric.identity(3)
[[1,0,0],
[0,1,0],
[0,0,1]]
</pre>
Random Arrays can also be created:
<pre >
> numeric.random([2,3])
[[0.05303,0.1537,0.7280],
[0.3839,0.08818,0.6316]]
</pre>
You can generate a vector of evenly spaced values:
<pre>
> numeric.linspace(1,5);
[1,2,3,4,5]
> numeric.linspace(1,3,5);
[1,1.5,2,2.5,3]
</pre>
<h1>Arithmetic operations</h1>
The standard arithmetic operations have been vectorized:
<pre>
> numeric.addVV([1,2],[3,4])
[4,6]
> numeric.addVS([1,2],3)
[4,5]
</pre>
There are also polymorphic functions:
<pre>
> numeric.add(1,[2,3])
[3,4]
> numeric.add([1,2,3],[4,5,6])
[5,7,9]
</pre>
The other arithmetic operations are available:
<pre>
> numeric.sub([1,2],[3,4])
[-2,-2]
> numeric.mul([1,2],[3,4])
[3,8]
> numeric.div([1,2],[3,4])
[0.3333,0.5]
</pre>
The in-place operators (such as +=) are also available:
<pre>
> v = [1,2,3,4]; numeric.addeq(v,3); v
[4,5,6,7]
> numeric.subeq([1,2,3],[5,3,1])
[-4,-1,2]
</pre>
Unary operators:
<pre>
> numeric.neg([1,-2,3])
[-1,2,-3]
> numeric.isFinite([10,NaN,Infinity])
[true,false,false]
> numeric.isNaN([10,NaN,Infinity])
[false,true,false]
</pre>
<!--
<pre>
> n = 41; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMsmall(numeric.inv(A),A),numeric.identity(n)))<1e-12
true
> n = 42; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMsmall(numeric.inv(A),A),numeric.identity(n)))<1e-12
true
> n = 43; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMsmall(numeric.inv(A),A),numeric.identity(n)))<1e-12
true
> n = 44; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMbig(numeric.inv(A),A),numeric.identity(n)))<1e-12
true
> n = 45; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMbig(numeric.inv(A),A),numeric.identity(n)))<1e-12
true
> n = 46; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMbig(numeric.inv(A),A),numeric.identity(n)))<1e-12
true
</pre>
-->
<h1>Linear algebra</h1>
Matrix products are implemented in the functions
<tt>numeric.dotVV()</tt>
<tt>numeric.dotVM()</tt>
<tt>numeric.dotMV()</tt>
<tt>numeric.dotMM()</tt>:
<pre>
> numeric.dotVV([1,2],[3,4])
11
> numeric.dotVM([1,2],[[3,4],[5,6]])
[13,16]
> numeric.dotMV([[1,2],[3,4]],[5,6])
[17,39]
> numeric.dotMMbig([[1,2],[3,4]],[[5,6],[7,8]])
[[19,22],
[43,50]]
> numeric.dotMMsmall([[1,2],[3,4]],[[5,6],[7,8]])
[[19,22],
[43,50]]
> numeric.dot([1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9])
285
</pre>
The function <tt>numeric.dot()</tt> is "polymorphic" and selects the appropriate Matrix product:
<pre>
> numeric.dot([1,2,3],[4,5,6])
32
> numeric.dot([[1,2,3],[4,5,6]],[7,8,9])
[50,122]
</pre>
The determinant:
<pre>
> numeric.det([[1,2],[3,4]]);
-2
> numeric.det([[6,8,4,2,8,5],[3,5,2,4,9,2],[7,6,8,3,4,5],[5,5,2,8,1,6],[3,2,2,4,2,2],[8,3,2,2,4,1]]);
-1404
</pre>
The matrix inverse:
<pre>
> numeric.inv([[1,2],[3,4]])
[[ -2, 1],
[ 1.5, -0.5]]
</pre>
The transpose:
<pre>
> numeric.transpose([[1,2,3],[4,5,6]])
[[1,4],
[2,5],
[3,6]]
> numeric.transpose([[1,2,3,4,5,6,7,8,9,10,11,12]])
[[ 1],
[ 2],
[ 3],
[ 4],
[ 5],
[ 6],
[ 7],
[ 8],
[ 9],
[10],
[11],
[12]]
</pre>
You can compute the 2-norm of an Array, which is the square root of the sum of the squares of the entries.
<pre>
> numeric.norm2([1,2])
2.236
</pre>
Computing the tensor product of two vectors:
<pre>
> numeric.tensor([1,2],[3,4,5])
[[3,4,5],
[6,8,10]]
</pre>
<h1>Data manipulation</h1>
There are also some data manipulation functions. You can parse dates:
<pre>
> numeric.parseDate(['1/13/2013','2001-5-9, 9:31']);
[1.358e12,9.894e11]
</pre>
Parse floating point quantities:
<pre>
> numeric.parseFloat(['12','0.1'])
[12,0.1]
</pre>
Parse CSV files:
<pre>
> numeric.parseCSV('a,b,c\n1,2.3,.3\n4e6,-5.3e-8,6.28e+4')
[[ "a", "b", "c"],
[ 1, 2.3, 0.3],
[ 4e6, -5.3e-8, 62800]]
> numeric.toCSV([[1.23456789123,2],[3,4]])
"1.23456789123,2
3,4
"
</pre>
You can also fetch a URL (a thin wrapper around XMLHttpRequest):
<pre>
> numeric.getURL('tools/helloworld.txt').responseText
"Hello, world!"
</pre>
<h1>Complex linear algebra</h1>
You can also manipulate complex numbers:
<pre>
> z = new numeric.T(3,4);
{x: 3, y: 4}
> z.add(5)
{x: 8, y: 4}
> w = new numeric.T(2,8);
{x: 2, y: 8}
> z.add(w)
{x: 5, y: 12}
> z.mul(w)
{x: -26, y: 32}
> z.div(w)
{x:0.5588,y:-0.2353}
> z.sub(w)
{x:1, y:-4}
</pre>
Complex vectors and matrices can also be handled:
<pre>
> z = new numeric.T([1,2],[3,4]);
{x: [1,2], y: [3,4]}
> z.abs()
{x:[3.162,4.472],y:}
> z.conj()
{x:[1,2],y:[-3,-4]}
> z.norm2()
5.477
> z.exp()
{x:[-2.691,-4.83],y:[0.3836,-5.592]}
> z.cos()
{x:[-1.528,-2.459],y:[0.1658,-2.745]}
> z.sin()
{x:[0.2178,-2.847],y:[1.163,2.371]}
> z.log()
{x:[1.151,1.498],y:[1.249,1.107]}
</pre>
Complex matrices:
<pre>
> A = new numeric.T([[1,2],[3,4]],[[0,1],[2,-1]]);
{x:[[1, 2],
[3, 4]],
y:[[0, 1],
[2,-1]]}
> A.inv();
{x:[[0.125,0.125],
[ 0.25, 0]],
y:[[ 0.5,-0.25],
[-0.375,0.125]]}
> A.inv().dot(A)
{x:[[1, 0],
[0, 1]],
y:[[0,-2.776e-17],
[0, 0]]}
> A.get([1,1])
{x: 4, y: -1}
> A.transpose()
{ x: [[1, 3],
[2, 4]],
y: [[0, 2],
[1,-1]] }
> A.transjugate()
{ x: [[ 1, 3],
[ 2, 4]],
y: [[ 0,-2],
[-1, 1]] }
> numeric.T.rep([2,2],new numeric.T(2,3));
{ x: [[2,2],
[2,2]],
y: [[3,3],
[3,3]] }
</pre>
<h1>Eigenvalues</h1>
Eigenvalues:
<pre>
> A = [[1,2,5],[3,5,0],[7,-3,5]];
[[ 1, 2, 5],
[ 3, 5, 0],
[ 7,-3, 5]]
> B = numeric.eig(A);
{lambda:{x:[-4.068,8.742,6.326],y:},
E: {x: [[ 0.7281,-0.5634, 0.4034 ],
[-0.2409,-0.4516, 0.9127 ],
[-0.6417,-0.6918, 0.06462]],
y:}}
> C = B.E.dot(numeric.T.diag(B.lambda)).dot(B.E.inv());
{x:[[ 1, 2, 5 ],
[ 3, 5,-3.109e-15],
[ 7, -3, 5 ]],
y:}
> C.sub(A).norm2()<1e-13
true
</pre>
Note that eigenvalues and eigenvectors are returned as complex numbers (type <tt>numeric.T</tt>). This is because
eigenvalues are often complex even when the matrix is real.<br><br>
<h1>Singular value decomposition (Shanti Rao)</h1>
Shanti Rao kindly emailed me an implementation of the Golub and Reisch algorithm:
<pre>
> A = [[22,10,2,3,7],[14,7,10,0,8],[-1,13,-1,-11,3],[-3,-2,13,-2,4],[9,8,1,-2,4],[9,1,-7,5,-1],[2,-6,6,5,1],[4,5,0,-2,2]]
[[ 22, 10, 2, 3, 7],
[ 14, 7, 10, 0, 8],
[ -1, 13, -1,-11, 3],
[ -3, -2, 13, -2, 4],
[ 9, 8, 1, -2, 4],
[ 9, 1, -7, 5, -1],
[ 2, -6, 6, 5, 1],
[ 4, 5, 0, -2, 2]]
> numeric.svd(A)
{U:
[[ -0.7071, -0.1581, 0.1768, 0.2494, 0.4625],
[ -0.5303, -0.1581, -0.3536, 0.1556, -0.4984],
[ -0.1768, 0.7906, -0.1768, -0.1546, 0.3967],
[ -1.506e-17, -0.1581, -0.7071, -0.3277, 0.1],
[ -0.3536, 0.1581, 1.954e-15, -0.07265, -0.2084],
[ -0.1768, -0.1581, 0.5303, -0.5726, -0.05555],
[ -7.109e-18, -0.4743, -0.1768, -0.3142, 0.4959],
[ -0.1768, 0.1581, 1.915e-15, -0.592, -0.2791]],
S:
[ 35.33, 20, 19.6, 0, 0],
V:
[[ -0.8006, -0.3162, 0.2887, -0.4191, 0],
[ -0.4804, 0.6325, 7.768e-15, 0.4405, 0.4185],
[ -0.1601, -0.3162, -0.866, -0.052, 0.3488],
[ 4.684e-17, -0.6325, 0.2887, 0.6761, 0.2442],
[ -0.3203, 3.594e-15, -0.2887, 0.413, -0.8022]]}
</pre>
<!--
Some further tests.
<pre>
> n = 31; A = numeric.random([n,n]); B = numeric.eig(A); B.E.dot(numeric.T.diag(B.lambda).dot(B.E.inv())).sub(A).norm2()<1e-12
true
> n = 32; A = numeric.random([n,n]); B = numeric.eig(A); B.E.dot(numeric.T.diag(B.lambda)).dot(B.E.inv()).sub(A).norm2()<1e-12
true
> n = 33; A = numeric.random([n,n]); B = numeric.eig(A); B.E.dot(numeric.T.diag(B.lambda)).dot(B.E.inv()).sub(A).norm2()<1e-12
true
> n = 34; A = numeric.random([n,n]); B = numeric.eig(A); B.E.dot(numeric.T.diag(B.lambda)).dot(B.E.inv()).sub(A).norm2()<1e-12
true
> n = 41; A = numeric.random([n,n]); B = numeric.eig(A); B.E.dot(numeric.T.diag(B.lambda)).dot(B.E.inv()).sub(A).norm2()<1e-12
true
> n = 42; A = numeric.random([n,n]); B = numeric.eig(A); B.E.dot(numeric.T.diag(B.lambda)).dot(B.E.inv()).sub(A).norm2()<1e-12
true
> n = 43; A = numeric.random([n,n]); B = numeric.eig(A); B.E.dot(numeric.T.diag(B.lambda)).dot(B.E.inv()).sub(A).norm2()<1e-12
true
> n = 44; A = numeric.random([n,n]); B = numeric.eig(A); B.E.dot(numeric.T.diag(B.lambda)).dot(B.E.inv()).sub(A).norm2()<1e-12
true
> m = 17; n = 12; A = numeric.random([m,n]); B = numeric.svd(A); U = new numeric.T(B.U); V = new numeric.T(B.V); U.dot(numeric.T.diag(B.S)).dot(V.transpose()).sub(A).norm2()<1e-12
true
> m = 21; n = 19; A = numeric.random([m,n]); B = numeric.svd(A); U = new numeric.T(B.U); V = new numeric.T(B.V); U.dot(numeric.T.diag(B.S)).dot(V.transpose()).sub(A).norm2()<1e-12
true
> m = 33; n = 33; A = numeric.random([m,n]); B = numeric.svd(A); U = new numeric.T(B.U); V = new numeric.T(B.V); U.dot(numeric.T.diag(B.S)).dot(V.transpose()).sub(A).norm2()<1e-12
true
> m = 59; n = 42; A = numeric.random([m,n]); B = numeric.svd(A); U = new numeric.T(B.U); V = new numeric.T(B.V); U.dot(numeric.T.diag(B.S)).dot(V.transpose()).sub(A).norm2()<1e-12
true
</pre>
-->
<h1>Sparse linear algebra</h1>
Sparse linear algebra:
<pre>
> numeric.sidentity(3)
[[1],
[ ,1],
[ , ,1]]
> numeric.stranspose([[1],[2,3],[4,5,6]])
[[1,2,4],
[ ,3,5],
[ , ,6]]
> A = [[2,-1],[-1,2,-1],[,-1,2]]; lup = numeric.sLUP(A)
{L: [[ 1],
[-0.5, 1],
[ ,-0.6667, 1]],
U: [[ 2, -1],
[ , 1.5, -1],
[ , , 1.333]],
P: [0,1,2],
Pinv: [0,1,2]}
> numeric.sdot(lup.L,lup.U)
[[ 2, -1],
[ -1, 2, -1],
[ , -1, 2]]
> x = [3,1,7]; b = numeric.sdot(A,x);
[5,-8,13]
> numeric.sLUPsolve(lup,b)
[3,1,7]
</pre>
<!--
Some more tests.
<pre>
> numeric.sdot([1,2,3],[,4,5])
23
> numeric.sdot([1, ,3],[[4,5,],[,6,7],[1,,8]])
[7,5,24]
> numeric.sdot([[3,1],[4,5,9],[,3,2]],[7,3])
[24,43,9]
</pre>
-->
The <tt>numeric.sscatter()</tt> and <tt>numeric.sgather()</tt> functions can be used to convert between
sparse matrices and the coordinate encoding:
<pre>
> A = numeric.sscatter([[0,0,1,1,1,2,2],[0,1,0,1,2,1,2],[1,2,3,4,5,6,7]])
[[1,2],
[3,4,5],
[ ,6,7]]
> numeric.sgather(A)
[[0,0,1,1,1,2,2],
[0,1,0,1,2,1,2],
[1,2,3,4,5,6,7]]
</pre>
<h1>Coordinate matrices</h1>
Sparse matrices can be very slow in certain browsers. Functions that
work with matrices in the coordinate encoding using band strategies may execute faster if
your matrices happen to be banded.<br><br>
LU decomposition:
<pre>
> lu = numeric.cLU([[0,0,1,1,1,2,2],[0,1,0,1,2,1,2],[2,-1,-1,2,-1,-1,2]])
{U:[[ 0, 0, 1, 1, 2 ],
[ 0, 1, 1, 2, 2 ],
[ 2, -1, 1.5, -1, 1.333]],
L:[[ 0, 1, 1, 2, 2 ],
[ 0, 0, 1, 1, 2 ],
[ 1, -0.5, 1,-0.6667, 1 ]]}
> numeric.cLUsolve(lu,[5,-8,13])
[3,1,7]
</pre>
Note that <tt>numeric.cLU()</tt> does not have any pivoting.
<h1>Solving PDEs</h1>
The functions <tt>numeric.cgrid()</tt> and <tt>numeric.cdelsq()</tt> can be used to obtain a
numerical Laplacian for a domain.
<pre>
> g = numeric.cgrid(5)
[[-1,-1,-1,-1,-1],
[-1, 0, 1, 2,-1],
[-1, 3, 4, 5,-1],
[-1, 6, 7, 8,-1],
[-1,-1,-1,-1,-1]]
> coordL = numeric.cdelsq(g)
[[ 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8],
[ 1, 3, 0, 0, 2, 4, 1, 1, 5, 2, 0, 4, 6, 3, 1, 3, 5, 7, 4, 2, 4, 8, 5, 3, 7, 6, 4, 6, 8, 7, 5, 7, 8],
[-1,-1, 4,-1,-1,-1, 4,-1,-1, 4,-1,-1,-1, 4,-1,-1,-1,-1, 4,-1,-1,-1, 4,-1,-1, 4,-1,-1,-1, 4,-1,-1, 4]]
> L = numeric.sscatter(coordL); // Just to see what it looks like
[[ 4, -1, , -1],
[ -1, 4, -1, , -1],
[ , -1, 4, , , -1],
[ -1, , , 4, -1, , -1],
[ , -1, , -1, 4, -1, , -1],
[ , , -1, , -1, 4, , , -1],
[ , , , -1, , , 4, -1],
[ , , , , -1, , -1, 4, -1],
[ , , , , , -1, , -1, 4]]
> lu = numeric.cLU(coordL); x = numeric.cLUsolve(lu,[1,1,1,1,1,1,1,1,1]);
[0.6875,0.875,0.6875,0.875,1.125,0.875,0.6875,0.875,0.6875]
> numeric.cdotMV(coordL,x)
[1,1,1,1,1,1,1,1,1]
> G = numeric.rep([5,5],0); for(i=0;i<5;i++) for(j=0;j<5;j++) if(g[i][j]>=0) G[i][j] = x[g[i][j]]; G
[[ 0 , 0 , 0 , 0 , 0 ],
[ 0 , 0.6875, 0.875 , 0.6875, 0 ],
[ 0 , 0.875 , 1.125 , 0.875 , 0 ],
[ 0 , 0.6875, 0.875 , 0.6875, 0 ],
[ 0 , 0 , 0 , 0 , 0 ]]
> workshop.html('<img src="'+numeric.imageURL(numeric.mul([G,G,G],200))+'" width=100 />');
<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAUAAAAFCAIAAAACDbGyAAAAcElEQVQIHQARAO7/AAAAAAAAAAAAAAAAAAAAAAAAEADv/wAAAIqKiq+vr4mJiQAAAAAAEADv/wAAAK+vr+Hh4a+vrwAAAAAAEADv/wAAAIqKiq+vr4qKigAAAAABEADv/wAAAAAAAAAAAAAAAAAAAACRjRFNqL3leAAAAABJRU5ErkJggg==" width=100 />
</pre>
You can also work on an L-shaped or arbitrary-shape domain:
<pre>
> numeric.cgrid(6,'L')
[[-1,-1,-1,-1,-1,-1],
[-1, 0, 1,-1,-1,-1],
[-1, 2, 3,-1,-1,-1],
[-1, 4, 5, 6, 7,-1],
[-1, 8, 9,10,11,-1],
[-1,-1,-1,-1,-1,-1]]
> numeric.cgrid(5,function(i,j) { return i!==2 || j!==2; })
[[-1,-1,-1,-1,-1],
[-1, 0, 1, 2,-1],
[-1, 3,-1, 4,-1],
[-1, 5, 6, 7,-1],
[-1,-1,-1,-1,-1]]
</pre>
<h1>Cubic splines</h1>
You can do some (natural) cubic spline interpolation:
<pre>
> numeric.spline([1,2,3,4,5],[1,2,1,3,2]).at(numeric.linspace(1,5,10))
[ 1, 1.731, 2.039, 1.604, 1.019, 1.294, 2.364, 3.085, 2.82, 2]
</pre>
For clamped splines:
<pre>
> numeric.spline([1,2,3,4,5],[1,2,1,3,2],0,0).at(numeric.linspace(1,5,10))
[ 1, 1.435, 1.98, 1.669, 1.034, 1.316, 2.442, 3.017, 2.482, 2]
</pre>
For periodic splines:
<pre>
> numeric.spline([1,2,3,4],[0.8415,0.04718,-0.8887,0.8415],'periodic').at(numeric.linspace(1,4,10))
[ 0.8415, 0.9024, 0.5788, 0.04718, -0.5106, -0.8919, -0.8887, -0.3918, 0.3131, 0.8415]
</pre>
Vector splines:
<pre>
> numeric.spline([1,2,3],[[0,1],[1,0],[0,1]]).at(1.78)
[0.9327,0.06728]
</pre>
Spline differentiation:
<pre>
> xs = [0,1,2,3]; numeric.spline(xs,numeric.sin(xs)).diff().at(1.5)
0.07302
</pre>
Find all the roots:
<pre>
> xs = numeric.linspace(0,30,31); ys = numeric.sin(xs); s = numeric.spline(xs,ys).roots();
[0, 3.142, 6.284, 9.425, 12.57, 15.71, 18.85, 21.99, 25.13, 28.27]
</pre>
<h1>Fast Fourier Transforms</h1>
FFT on numeric.T objects:
<pre>
> z = (new numeric.T([1,2,3,4,5],[6,7,8,9,10])).fft()
{x:[15,-5.941,-3.312,-1.688, 0.941],
y:[40, 0.941,-1.688,-3.312,-5.941]}
</pre>
Also inverse FFT:
<pre>
> z.ifft()
{x:[1,2,3,4,5],
y:[6,7,8,9,10]}
</pre>
<h1>Quadratic Programming (Alberto Santini)</h1>
The Quadratic Programming function <tt>numeric.solveQP()</tt> is based on <a href="https://github.com/albertosantini/node-quadprog">Alberto Santini's
quadprog</a>, which is itself a port of the corresponding
R routines.
<pre>
> numeric.solveQP([[1,0,0],[0,1,0],[0,0,1]],[0,5,0],[[-4,2,0],[-3,1,-2],[0,0,1]],[-8,2,0]);
{ solution: [0.4762,1.048,2.095],
value: [-2.381],
unconstrained_solution:[ 0, 5, 0],
iterations: [ 3, 0],
iact: [ 3, 2, 0],
message: "" }
</pre>
<h1>Unconstrained optimization</h1>
We also include a simple unconstrained optimization routine. Here are some demos from from Moré et al., 1981:
<pre>
> sqr = function(x) { return x*x; }; numeric.uncmin(function(x) { return sqr(10*(x[1]-x[0]*x[0])) + sqr(1-x[0]); },[-1.2,1]).solution
[1,1]
> f = function(x) { return sqr(-13+x[0]+((5-x[1])*x[1]-2)*x[1])+sqr(-29+x[0]+((x[1]+1)*x[1]-14)*x[1]); }; x0 = numeric.uncmin(f,[0.5,-2]).solution
[11.41,-0.8968]
> f = function(x) { return sqr(1e4*x[0]*x[1]-1)+sqr(Math.exp(-x[0])+Math.exp(-x[1])-1.0001); }; x0 = numeric.uncmin(f,[0,1]).solution
[1.098e-5,9.106]
> f = function(x) { return sqr(x[0]-1e6)+sqr(x[1]-2e-6)+sqr(x[0]*x[1]-2)}; x0 = numeric.uncmin(f,[0,1]).solution
[1e6,2e-6]
> f = function(x) { return sqr(1.5-x[0]*(1-x[1]))+sqr(2.25-x[0]*(1-x[1]*x[1]))+sqr(2.625-x[0]*(1-x[1]*x[1]*x[1])); }; x0 = numeric.uncmin(f,[1,1]).solution
[3,0.5]
> f = function(x) { var ret = 0,i; for(i=1;i<=10;i++) ret+=sqr(2+2*i-Math.exp(i*x[0])-Math.exp(i*x[1])); return ret; }; x0 = numeric.uncmin(f,[0.3,0.4]).solution
[0.2578,0.2578]
> y = [0.14,0.18,0.22,0.25,0.29,0.32,0.35,0.39,0.37,0.58,0.73,0.96,1.34,2.10,4.39]; f = function(x) { var ret = 0,i; for(i=1;i<=15;i++) ret+=sqr(y[i-1]-(x[0]+i/((16-i)*x[1]+Math.min(i,16-i)*x[2]))); return ret; }; x0 = numeric.uncmin(f,[1,1,1]).solution
[0.08241,1.133,2.344]
> y = [0.0009,0.0044,0.0175,0.0540,0.1295,0.2420,0.3521,0.3989,0.3521,0.2420,0.1295,0.0540,0.0175,0.0044,0.0009]; f = function(x) { var ret = 0,i; for(i=1;i<=15;i++) ret+=sqr(x[0]*Math.exp(-x[1]*sqr((8-i)/2-x[2])/2)-y[i-1]); return ret; }; x0 = numeric.div(numeric.round(numeric.mul(numeric.uncmin(f,[1,1,1]).solution,1000)),1000)
[0.399,1,0]
> f = function(x) { return sqr(x[0]+10*x[1])+5*sqr(x[2]-x[3])+sqr(sqr(x[1]-2*x[2]))+10*sqr(x[0]-x[3]); }; x0 = numeric.div(numeric.round(numeric.mul(numeric.uncmin(f,[3,-1,0,1]).solution,1e5)),1e5)
[0,0,0,0]
> f = function(x) { return sqr(10*(x[1]-x[0]*x[0]))+sqr(1-x[0])+90*sqr(x[3]-x[2]*x[2])+sqr(1-x[2])+10*sqr(x[1]+x[3]-2)+0.1*sqr(x[1]-x[3]); }; x0 = numeric.uncmin(f,[-3,-1,-3,-1]).solution
[1,1,1,1]
> y = [0.1957,0.1947,0.1735,0.1600,0.0844,0.0627,0.0456,0.0342,0.0323,0.0235,0.0246]; u = [4,2,1,0.5,0.25,0.167,0.125,0.1,0.0833,0.0714,0.0625]; f = function(x) { var ret=0, i; for(i=0;i<11;++i) ret += sqr(y[i]-x[0]*(u[i]*u[i]+u[i]*x[1])/(u[i]*u[i]+u[i]*x[2]+x[3])); return ret; }; x0 = numeric.uncmin(f,[0.25,0.39,0.415,0.39]).solution
[ 0.1928, 0.1913, 0.1231, 0.1361]
> y = [0.844,0.908,0.932,0.936,0.925,0.908,0.881,0.850,0.818,0.784,0.751,0.718,0.685,0.658,0.628,0.603,0.580,0.558,0.538,0.522,0.506,0.490,0.478,0.467,0.457,0.448,0.438,0.431,0.424,0.420,0.414,0.411,0.406]; f = function(x) { var ret=0, i; for(i=0;i<33;++i) ret += sqr(y[i]-(x[0]+x[1]*Math.exp(-10*i*x[3])+x[2]*Math.exp(-10*i*x[4]))); return ret; }; x0 = numeric.uncmin(f,[0.5,1.5,-1,0.01,0.02]).solution
[ 0.3754, 1.936, -1.465, 0.01287, 0.02212]
> f = function(x) { var ret=0, i,ti,yi,exp=Math.exp; for(i=1;i<=13;++i) { ti = 0.1*i; yi = exp(-ti)-5*exp(-10*ti)+3*exp(-4*ti); ret += sqr(x[2]*exp(-ti*x[0])-x[3]*exp(-ti*x[1])+x[5]*exp(-ti*x[4])-yi); } return ret; }; x0 = numeric.uncmin(f,[1,2,1,1,1,1],1e-14).solution; f(x0)<1e-20;
true
</pre>
There are optional parameters to <tt>numeric.uncmin(f,x0,tol,gradient,maxit,callback)</tt>. The iteration stops when
the gradient or step size is less than the optional parameter <tt>tol</tt>. The <tt>gradient()</tt> parameter is a function that computes the
gradient of <tt>f()</tt>. If it is not provided, a numerical gradient is used. The iteration stops when
<tt>maxit</tt> iterations have been performed. The optional <tt>callback()</tt> parameter, if provided, is called at each step:
<pre>
> z = []; cb = function(i,x,f,g,H) { z.push({i:i, x:x, f:f, g:g, H:H }); }; x0 = numeric.uncmin(function(x) { return Math.cos(2*x[0]); },[1],1e-10,function(x) { return [-2*Math.sin(2*x[0])]; },100,cb);
{solution: [1.571],
f: -1,
gradient: [2.242e-11],
invHessian: [[0.25]],
iterations: 6,
message: "Newton step smaller than tol"}
> z
[{i:0, x:[1 ], f:-0.4161, g: [-1.819 ] , H:[[1 ]] },
{i:2, x:[1.909], f:-0.7795, g: [ 1.253 ] , H:[[0.296 ]] },
{i:3, x:[1.538], f:-0.9979, g: [-0.1296 ] , H:[[0.2683]] },
{i:4, x:[1.573], f:-1 , g: [ 9.392e-3] , H:[[0.2502]] },
{i:5, x:[1.571], f:-1 , g: [-6.105e-6] , H:[[0.25 ]] },
{i:6, x:[1.571], f:-1 , g: [ 2.242e-11] , H:[[0.25 ]] }]
</pre>
<h1>Solving ODEs</h1>
The function <tt>numeric.dopri()</tt> is an implementation of the Dormand-Prince-Runge-Kutta integrator with
adaptive time-stepping:
<pre>
> sol = numeric.dopri(0,1,1,function(t,y) { return y; })
{ x: [ 0, 0.1, 0.1522, 0.361, 0.5792, 0.7843, 0.9813, 1],
y: [ 1, 1.105, 1.164, 1.435, 1.785, 2.191, 2.668, 2.718],
f: [ 1, 1.105, 1.164, 1.435, 1.785, 2.191, 2.668, 2.718],
ymid: [ 1.051, 1.134, 1.293, 1.6, 1.977, 2.418, 2.693],
iterations:8,
events:,
message:""}
</pre>
The return value <tt>sol</tt> contains the x and y values of the solution.
If you need to know the value of the solution between the given x values, use the function
<tt>sol.at()</tt>, which uses the extra information contained in <tt>sol.ymid</tt> and <tt>sol.f</tt> to
produce approximations between these points: