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<!doctype html>
<html>
<head>
<link rel="SHORTCUT ICON" href="favicon.ico">
<link href='http://fonts.googleapis.com/css?family=Lato' rel='stylesheet' type='text/css'>
<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>
IN> numeric.seedrandom.seedrandom('1'); Math.random = numeric.seedrandom.random; Math.random();
OUT> 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>ccsDim</tt><td>Dimensions of sparse matrix
<tr><td><tt>ccsDot</tt><td>Sparse matrix-matrix product
<tr><td><tt>ccsFull</tt><td>Convert sparse to full
<tr><td><tt>ccsGather</tt><td>Gather entries of sparse matrix
<tr><td><tt>ccsGetBlock</tt><td>Get rows/columns of sparse matrix
<tr><td><tt>ccsLUP</tt><td>Compute LUP decomposition of sparse matrix
<tr><td><tt>ccsLUPSolve</tt><td>Solve Ax=b using LUP decomp
<tr><td><tt>ccsScatter</tt><td>Scatter entries of sparse matrix
<tr><td><tt>ccsSparse</tt><td>Convert from full to sparse
<tr><td><tt>ccsTSolve</tt><td>Solve upper/lower triangular system
<tr><td><tt>ccs<op></td><td>Supported ops include: add/div/mul/geq/etc...
<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
</table>
<td valign="top" width="33%">
<table>
<tr><td><b>Function</b><td><b>Description</b>
<tr><td colspan=2><hr>
<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
<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>LU</tt><td>Dense LU decomposition
<tr><td><tt>LUsolve</tt><td>Dense LU solve
<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
</table>
<td valign="top" width="33%">
<table>
<tr><td><b>Function</b><td><b>Description</b>
<tr><td colspan=2><hr>
<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>same</tt><td>x and y are entrywise identical
<tr><td><tt>seedrandom</tt><td>The seedrandom module
<tr><td><tt>setBlock</tt><td>Set a block of a matrix
<tr><td><tt>sin</tt><td>Pointwise Math.sin(x)
<tr><td><tt>solve</tt><td>Solve Ax=b
<tr><td><tt>solveLP</tt><td>Solve a linear programming problem
<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>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>
<br>
<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>
IN> A = [[1,2,3],[4,5,6]];
OUT> [[1,2,3],
[4,5,6]]
IN> x = [7,8,9]
OUT> [7,8,9]
IN> numeric.dot(A,x);
OUT> [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>
IN> x = [7,8,9];
y = [10,1,2];
numeric['+'](x,y)
OUT> [17,9,11]
IN> numeric['>'](x,y)
OUT> [false,true,true]
</pre>
These operators can also be called with plain Javascript function names:
<pre>
IN> numeric.add([7,8,9],[10,1,2])
OUT> [17,9,11]
</pre>
You can also use these operators with three or more parameters:
<pre>
IN> numeric.add([1,2],[3,4],[5,6],[7,8])
OUT> [16,20]
</pre>
The function <tt>numeric.inv()</tt> can be used to compute the inverse of an invertible matrix:
<pre>
IN> A = [[1,2,3],[4,5,6],[7,1,9]]
OUT> [[1,2,3],
[4,5,6],
[7,1,9]]
IN> Ainv = numeric.inv(A);
OUT> [[-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>
IN> numeric.precision = 10; x = 3.141592653589793
OUT> 3.141592654
IN> numeric.precision = 4; x
OUT> 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>
IN> numeric.identity(100)
OUT> ...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>
IN> numeric.largeArray = 2; A = numeric.identity(4)
OUT> ...Large Array...
IN> numeric.largeArray = 50; A
OUT> [[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>
IN> numeric.exp([1,2]);
OUT> [2.718,7.389]
IN> numeric.exp([[1,2],[3,4]])
OUT> [[2.718, 7.389],
[20.09, 54.6]]
IN> numeric.abs([-2,3])
OUT> [2,3]
IN> numeric.acos([0.1,0.2])
OUT> [1.471,1.369]
IN> numeric.asin([0.1,0.2])
OUT> [0.1002,0.2014]
IN> numeric.atan([1,2])
OUT> [0.7854,1.107]
IN> numeric.atan2([1,2],[3,4])
OUT> [0.3218,0.4636]
IN> numeric.ceil([-2.2,3.3])
OUT> [-2,4]
IN> numeric.floor([-2.2,3.3])
OUT> [-3,3]
IN> numeric.log([1,2])
OUT> [0,0.6931]
IN> numeric.pow([2,3],[0.25,7.1])
OUT> [1.189,2441]
IN> numeric.round([-2.2,3.3])
OUT> [-2,3]
IN> numeric.sin([1,2])
OUT> [0.8415,0.9093]
IN> numeric.sqrt([1,2])
OUT> [1,1.414]
IN> numeric.tan([1,2])
OUT> [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>
IN> numeric.dim([1,2])
OUT> [2]
IN> numeric.dim([[1,2,3],[4,5,6]])
OUT> [2,3]
</pre>
You can perform a deep comparison of Arrays using <tt>numeric.same()</tt>:
<pre>
IN> numeric.same([1,2],[1,2])
OUT> true
IN> numeric.same([1,2],[1,2,3])
OUT> false
IN> numeric.same([1,2],[[1],[2]])
OUT> false
IN> numeric.same([[1,2],[3,4]],[[1,2],[3,4]])
OUT> true
IN> numeric.same([[1,2],[3,4]],[[1,2],[3,5]])
OUT> false
IN> numeric.same([[1,2],[2,4]],[[1,2],[3,4]])
OUT> false
</pre>
You can create a multidimensional Array from a given value using <tt>numeric.rep()</tt>
<pre>
IN> numeric.rep([3],5)
OUT> [5,5,5]
IN> numeric.rep([2,3],0)
OUT> [[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>
IN> sum = numeric.mapreduce('accum += xi','0'); sum([1,2,3])
OUT> 6
IN> sum([[1,2,3],[4,5,6]])
OUT> 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>
IN> numeric.any([false,true])
OUT> true
IN> numeric.any([[0,0,3.14],[0,false,0]])
OUT> true
IN> numeric.any([0,0,false])
OUT> false
IN> numeric.all([false,true])
OUT> false
IN> numeric.all([[1,4,3.14],["no",true,-1]])
OUT> true
IN> numeric.all([0,0,false])
OUT> false
</pre>
You can create a diagonal matrix using <tt>numeric.diag()</tt>
<pre>
IN> numeric.diag([1,2,3])
OUT> [[1,0,0],
[0,2,0],
[0,0,3]]
</pre>
The function <tt>numeric.identity()</tt> returns the identity matrix.
<pre>
IN> numeric.identity(3)
OUT> [[1,0,0],
[0,1,0],
[0,0,1]]
</pre>
Random Arrays can also be created:
<pre >
IN> numeric.random([2,3])
OUT> [[0.05303,0.1537,0.7280],
[0.3839,0.08818,0.6316]]
</pre>
You can generate a vector of evenly spaced values:
<pre>
IN> numeric.linspace(1,5);
OUT> [1,2,3,4,5]
IN> numeric.linspace(1,3,5);
OUT> [1,1.5,2,2.5,3]
</pre>
<!--
<pre>
IN> numeric.blockMatrix([[[[1,2],[3,4]],[[5,6],[7,8]]],
[[[11,12],[13,14]],[[15,16],[17,18]]]])
OUT> [[ 1, 2, 5, 6],
[ 3, 4, 7, 8],
[11,12,15,16],
[13,14,17,18]]
</pre>
-->
<h1>Arithmetic operations</h1>
The standard arithmetic operations have been vectorized:
<pre>
IN> numeric.addVV([1,2],[3,4])
OUT> [4,6]
IN> numeric.addVS([1,2],3)
OUT> [4,5]
</pre>
There are also polymorphic functions:
<pre>
IN> numeric.add(1,[2,3])
OUT> [3,4]
IN> numeric.add([1,2,3],[4,5,6])
OUT> [5,7,9]
</pre>
The other arithmetic operations are available:
<pre>
IN> numeric.sub([1,2],[3,4])
OUT> [-2,-2]
IN> numeric.mul([1,2],[3,4])
OUT> [3,8]
IN> numeric.div([1,2],[3,4])
OUT> [0.3333,0.5]
</pre>
The in-place operators (such as +=) are also available:
<pre>
IN> v = [1,2,3,4]; numeric.addeq(v,3); v
OUT> [4,5,6,7]
IN> numeric.subeq([1,2,3],[5,3,1])
OUT> [-4,-1,2]
</pre>
Unary operators:
<pre>
IN> numeric.neg([1,-2,3])
OUT> [-1,2,-3]
IN> numeric.isFinite([10,NaN,Infinity])
OUT> [true,false,false]
IN> numeric.isNaN([10,NaN,Infinity])
OUT> [false,true,false]
</pre>
<!--
<pre>
IN> n = 41; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMsmall(numeric.inv(A),A),numeric.identity(n)))<1e-12
OUT> true
IN> n = 42; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMsmall(numeric.inv(A),A),numeric.identity(n)))<1e-12
OUT> true
IN> n = 43; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMsmall(numeric.inv(A),A),numeric.identity(n)))<1e-12
OUT> true
IN> n = 44; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMbig(numeric.inv(A),A),numeric.identity(n)))<1e-12
OUT> true
IN> n = 45; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMbig(numeric.inv(A),A),numeric.identity(n)))<1e-12
OUT> true
IN> n = 46; A = numeric.random([n,n]); numeric.norm2(numeric.sub(numeric.dotMMbig(numeric.inv(A),A),numeric.identity(n)))<1e-12
OUT> 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>
IN> numeric.dotVV([1,2],[3,4])
OUT> 11
IN> numeric.dotVM([1,2],[[3,4],[5,6]])
OUT> [13,16]
IN> numeric.dotMV([[1,2],[3,4]],[5,6])
OUT> [17,39]
IN> numeric.dotMMbig([[1,2],[3,4]],[[5,6],[7,8]])
OUT> [[19,22],
[43,50]]
IN> numeric.dotMMsmall([[1,2],[3,4]],[[5,6],[7,8]])
OUT> [[19,22],
[43,50]]
IN> numeric.dot([1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,9])
OUT> 285
</pre>
The function <tt>numeric.dot()</tt> is "polymorphic" and selects the appropriate Matrix product:
<pre>
IN> numeric.dot([1,2,3],[4,5,6])
OUT> 32
IN> numeric.dot([[1,2,3],[4,5,6]],[7,8,9])
OUT> [50,122]
</pre>
Solving the linear problem Ax=b (<a href="https://github.com/yuzeh">Dan Huang</a>):
<pre>
IN> numeric.solve([[1,2],[3,4]],[17,39])
OUT> [5,6]
</pre>
The algorithm is based on the LU decomposition:
<pre>
IN> LU = numeric.LU([[1,2],[3,4]])
OUT> {LU:[[3 ,4 ],
[0.3333,0.6667]],
P:[1,1]}
IN> numeric.LUsolve(LU,[17,39])
OUT> [5,6]
</pre>
<!--
Stress testing
<pre>
IN> ns = [5,6,10,16,25,40,41];
for(j=0;j<ns.length;j++) {
n = ns[j];
for(k=0;k<10;++k) {
A = numeric.random([n,n]);
x = numeric.random([n]);
b = numeric.dot(A,x);
y = numeric.solve(A,b);
if(!(numeric.norminf(numeric.sub(x,y))<1e-10)) throw new Error(
"numeric.solve Stress Test:"+numeric.prettyPrint({j:j,k:k,A:A,x:x,b:b,y:y}));
}
}
"numeric.solve Stress Test OK";
OUT> "numeric.solve Stress Test OK"
</pre>
-->
The determinant:
<pre>
IN> numeric.det([[1,2],[3,4]]);
OUT> -2
IN> 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]]);
OUT> -1404
</pre>
The matrix inverse:
<pre>
IN> numeric.inv([[1,2],[3,4]])
OUT> [[ -2, 1],
[ 1.5, -0.5]]
</pre>
The transpose:
<pre>
IN> numeric.transpose([[1,2,3],[4,5,6]])
OUT> [[1,4],
[2,5],
[3,6]]
IN> numeric.transpose([[1,2,3,4,5,6,7,8,9,10,11,12]])
OUT> [[ 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>
IN> numeric.norm2([1,2])
OUT> 2.236
</pre>
Computing the tensor product of two vectors:
<pre>
IN> numeric.tensor([1,2],[3,4,5])
OUT> [[3,4,5],
[6,8,10]]
</pre>
<h1>Data manipulation</h1>
There are also some data manipulation functions. You can parse dates:
<pre>
IN> numeric.parseDate(['1/13/2013','2001-5-9, 9:31']);
OUT> [1.358e12,9.894e11]
</pre>
Parse floating point quantities:
<pre>
IN> numeric.parseFloat(['12','0.1'])
OUT> [12,0.1]
</pre>
Parse CSV files:
<pre>
IN> numeric.parseCSV('a,b,c\n1,2.3,.3\n4e6,-5.3e-8,6.28e+4')
OUT> [[ "a", "b", "c"],
[ 1, 2.3, 0.3],
[ 4e6, -5.3e-8, 62800]]
IN> numeric.toCSV([[1.23456789123,2],[3,4]])
OUT> "1.23456789123,2
3,4
"
</pre>
You can also fetch a URL (a thin wrapper around XMLHttpRequest):
<pre>
IN> numeric.getURL('tools/helloworld.txt').responseText
OUT> "Hello, world!"
</pre>
<h1>Complex linear algebra</h1>
You can also manipulate complex numbers:
<pre>
IN> z = new numeric.T(3,4);
OUT> {x: 3, y: 4}
IN> z.add(5)
OUT> {x: 8, y: 4}
IN> w = new numeric.T(2,8);
OUT> {x: 2, y: 8}
IN> z.add(w)
OUT> {x: 5, y: 12}
IN> z.mul(w)
OUT> {x: -26, y: 32}
IN> z.div(w)
OUT> {x:0.5588,y:-0.2353}
IN> z.sub(w)
OUT> {x:1, y:-4}
</pre>
Complex vectors and matrices can also be handled:
<pre>
IN> z = new numeric.T([1,2],[3,4]);
OUT> {x: [1,2], y: [3,4]}
IN> z.abs()
OUT> {x:[3.162,4.472],y:}
IN> z.conj()
OUT> {x:[1,2],y:[-3,-4]}
IN> z.norm2()
OUT> 5.477
IN> z.exp()
OUT> {x:[-2.691,-4.83],y:[0.3836,-5.592]}
IN> z.cos()
OUT> {x:[-1.528,-2.459],y:[0.1658,-2.745]}
IN> z.sin()
OUT> {x:[0.2178,-2.847],y:[1.163,2.371]}
IN> z.log()
OUT> {x:[1.151,1.498],y:[1.249,1.107]}
</pre>
Complex matrices:
<pre>
IN> A = new numeric.T([[1,2],[3,4]],[[0,1],[2,-1]]);
OUT> {x:[[1, 2],
[3, 4]],
y:[[0, 1],
[2,-1]]}
IN> A.inv();
OUT> {x:[[0.125,0.125],
[ 0.25, 0]],
y:[[ 0.5,-0.25],
[-0.375,0.125]]}
IN> A.inv().dot(A)
OUT> {x:[[1, 0],
[0, 1]],
y:[[0,-2.776e-17],
[0, 0]]}
IN> A.get([1,1])
OUT> {x: 4, y: -1}
IN> A.transpose()
OUT> { x: [[1, 3],
[2, 4]],
y: [[0, 2],
[1,-1]] }
IN> A.transjugate()
OUT> { x: [[ 1, 3],
[ 2, 4]],
y: [[ 0,-2],
[-1, 1]] }
IN> numeric.T.rep([2,2],new numeric.T(2,3));
OUT> { x: [[2,2],
[2,2]],
y: [[3,3],
[3,3]] }
</pre>
<h1>Eigenvalues</h1>
Eigenvalues:
<pre>
IN> A = [[1,2,5],[3,5,-1],[7,-3,5]];
OUT> [[ 1, 2, 5],
[ 3, 5, -1],
[ 7, -3, 5]]
IN> B = numeric.eig(A);
OUT> {lambda:{x:[-4.284,9.027,6.257],y:},
E:{x:[[ 0.7131,-0.5543,0.4019],
[-0.2987,-0.2131,0.9139],
[-0.6342,-0.8046,0.057 ]],
y:}}
IN> C = B.E.dot(numeric.T.diag(B.lambda)).dot(B.E.inv());
OUT> {x: [[ 1, 2, 5],
[ 3, 5, -1],
[ 7, -3, 5]],
y: }
</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>
IN> 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]];
numeric.svd(A);
OUT> {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>
IN> 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)
OUT> true
IN> 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)
OUT> true
IN> 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)
OUT> true
IN> 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)
OUT> true
IN> n = 35; 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)
OUT> true
IN> 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)
OUT> true
IN> 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)
OUT> true
IN> 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)
OUT> true
IN> 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)
OUT> true
IN> numeric.eig([[1, 0, 0], [0, 0.7181, -0.6960], [0, 0.6960, 0.7181]]) // This was a bug found by bdmartin
OUT> {lambda:
{x:
[ 1, 0.7181, 0.7181],
y:
[ 0, 0.696, -0.696]},
E:
{x:
[[ 1, 0, 0],
[ 0, 0, 0.7071],
[ 0, -0.7071, 0]],
y:
[[ 0, 0, 0],
[ 0, -0.7071, 0],
[ 0, 0, 0.7071]]}}
</pre>
-->
<h1>Sparse linear algebra</h1>
Sparse matrices are matrices that have a lot of zeroes. In numeric, sparse matrices are stored in the
"compressed column storage" ordering. Example:
<pre>
IN> A = [[1,2,0],
[0,3,0],
[2,0,5]];
SA = numeric.ccsSparse(A);
OUT> [[0,2,4,5],
[0,2,0,1,2],
[1,2,2,3,5]]
</pre>
The relation between A and its sparse representation SA is:
<pre >
A[i][SA[1][k]] = SA[2][k] with SA[0][i] ≤ k < SA[0][i+1]
</pre >
In other words, SA[2] stores the nonzero entries of A; SA[1] stores the row numbers and SA[0] stores the
offsets of the columns. See <i>I. DUFF, R. GRIMES, AND J. LEWIS, Sparse matrix test problems, ACM Trans. Math. Soft., 15 (1989), pp. 1-14.</i>
<pre>
IN> A = numeric.ccsSparse([[ 3, 5, 8,10, 8],
[ 7,10, 3, 5, 3],
[ 6, 3, 5, 1, 8],
[ 2, 6, 7, 1, 2],
[ 1, 2, 9, 3, 9]]);
OUT> [[0,5,10,15,20,25],
[0,1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,1,2,3,4],
[3,7,6,2,1,5,10,3,6,2,8,3,5,7,9,10,5,1,1,3,8,3,8,2,9]]
IN> numeric.ccsFull(A);
OUT> [[ 3, 5, 8,10, 8],
[ 7,10, 3, 5, 3],
[ 6, 3, 5, 1, 8],
[ 2, 6, 7, 1, 2],
[ 1, 2, 9, 3, 9]]
IN> numeric.ccsDot(numeric.ccsSparse([[1,2,3],[4,5,6]]),numeric.ccsSparse([[7,8],[9,10],[11,12]]))
OUT> [[0,2,4],
[0,1,0,1],
[58,139,64,154]]
IN> M = [[0,1,3,6],[0,0,1,0,1,2],[3,-1,2,3,-2,4]];
b = [9,3,2];
x = numeric.ccsTSolve(M,b);
OUT> [3.167,2,0.5]
IN> numeric.ccsDot(M,[[0,3],[0,1,2],x])
OUT> [[0,3],[0,1,2],[9,3,2]]
</pre>
We provide an LU=PA decomposition:
<pre>
IN> A = [[0,5,10,15,20,25],
[0,1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,1,2,3,4],
[3,7,6,2,1,5,10,3,6,2,8,3,5,7,9,10,5,1,1,3,8,3,8,2,9]];
LUP = numeric.ccsLUP(A);
OUT> {L:[[0,5,9,12,14,15],
[0,2,4,1,3,1,3,4,2,2,4,3,3,4,4],
[1,0.1429,0.2857,0.8571,0.4286,1,-0.1282,-0.5641,-0.1026,1,0.8517,0.7965,1,-0.67,1]],
U:[[0,1,3,6,10,15],
[0,0,1,0,1,2,0,1,2,3,0,1,2,3,4],
[7,10,-5.571,3,2.429,8.821,5,-3.286,1.949,5.884,3,5.429,9.128,0.1395,-3.476]],
P:[1,2,4,0,3],
Pinv:[3,0,1,4,2]}
IN> numeric.ccsFull(numeric.ccsDot(LUP.L,LUP.U))
OUT> [[ 7,10, 3, 5, 3],
[ 6, 3, 5, 1, 8],
[ 1, 2, 9, 3, 9],
[ 3, 5, 8,10, 8],
[ 2, 6, 7, 1, 2]]
IN> x = numeric.ccsLUPSolve(LUP,[96,63,82,51,89])
OUT> [3,1,4,1,5]
IN> X = numeric.trunc(numeric.ccsFull(numeric.ccsLUPSolve(LUP,A)),1e-15); // Solve LUX = PA
OUT> [[1,0,0,0,0],
[0,1,0,0,0],
[0,0,1,0,0],
[0,0,0,1,0],
[0,0,0,0,1]]
IN> numeric.ccsLUP(A,0.4).P;
OUT> [0,2,1,3,4]
</pre>
The LUP decomposition uses partial pivoting and has an optional thresholding argument.
With a threshold of 0.4, the pivots are [0,2,1,3,4] (only rows 1 and 2 have been exchanged) instead of the
"full partial pivoting" order above which was [1,2,4,0,3]. Threshold=0 gives no pivoting
and threshold=1 gives normal partial pivoting. Note that a small or zero threshold can result in numerical
instabilities and is normally used when the matrix A is already in some order that minimizes fill-in.
<!-- Stress test:
<pre>
IN> result = "Sparse LUP Stress Test OK";
for(k=0;k<1000;++k) {
A = numeric.ccsSparse(numeric.random([10,10]));
LUP = numeric.ccsLUP(A);
foo = numeric.ccsFull(numeric.ccsDot(LUP.L,LUP.U));
PA = numeric.ccsFull(numeric.ccsGetBlock(A,LUP.P));
res = numeric.norminf(numeric.sub(foo,PA));
if(!isFinite(res) || res>1e-6) {
result = {
code: "Failed during 1000 sparse LUP",
k:k,A:A,LUP:LUP,res:res
};
break;
};
};
result;
OUT> "Sparse LUP Stress Test OK"
</pre>
-->
We also support arithmetic on CCS matrices:
<pre>
IN> A = numeric.ccsSparse([[1,2,0],[0,3,0],[0,0,5]]);
B = numeric.ccsSparse([[2,9,0],[0,4,0],[-2,0,0]]);
numeric.ccsadd(A,B);
OUT> [[0,2,4,5],
[0,2,0,1,2],
[3,-2,11,7,5]]
</pre>
We also have scatter/gather functions
<pre>
IN> X = [[0,0,1,1,2,2],[0,1,1,2,2,3],[1,2,3,4,5,6]];
SX = numeric.ccsScatter(X);
OUT> [[0,1,3,5,6],
[0,0,1,1,2,2],
[1,2,3,4,5,6]]
IN> numeric.ccsGather(SX)
OUT> [[0,0,1,1,2,2],[0,1,1,2,2,3],[1,2,3,4,5,6]]
</pre>
<h1>Coordinate matrices</h1>
We also provide a banded matrix implementation using the coordinate encoding.<br><br>
LU decomposition:
<pre>
IN> lu = numeric.cLU([[0,0,1,1,1,2,2],[0,1,0,1,2,1,2],[2,-1,-1,2,-1,-1,2]])
OUT> {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 ]]}
IN> numeric.cLUsolve(lu,[5,-8,13])
OUT> [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>
IN> g = numeric.cgrid(5)
OUT>
[[-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]]
IN> coordL = numeric.cdelsq(g)
OUT>
[[ 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]]
IN> L = numeric.sscatter(coordL); // Just to see what it looks like
OUT>
[[ 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]]
IN> lu = numeric.cLU(coordL); x = numeric.cLUsolve(lu,[1,1,1,1,1,1,1,1,1]);
OUT> [0.6875,0.875,0.6875,0.875,1.125,0.875,0.6875,0.875,0.6875]
IN> numeric.cdotMV(coordL,x)
OUT> [1,1,1,1,1,1,1,1,1]
IN> 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
OUT>
[[ 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 ]]
IN> workshop.html('<img src="'+numeric.imageURL(numeric.mul([G,G,G],200))+'" width=100 />');
OUT>
<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>
IN> numeric.cgrid(6,'L')
OUT>
[[-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]]
IN> numeric.cgrid(5,function(i,j) { return i!==2 || j!==2; })
OUT>
[[-1,-1,-1,-1,-1],
[-1, 0, 1, 2,-1],
[-1, 3,-1, 4,-1],
[-1, 5, 6, 7,-1],