# linalg.js - Linear Algebra for the web.

Linear Algebra Expressions and operator overloading on the web.. or can you ?

Since every post should start with a nice image, here’s a rank 15,10,5 and 2 approximation of an image calculated in javascript :

And here’s the script that generates it :

var img = new Image();
img.src = '../res/testimage.jpg';
var U=Matrix(this), S, V, R;
"[U,S,V]=U.SVD"
[15,10,5,2].forEach(j=>{
for (var i=j; i < S.rows; i++) S[i]=0;
"R=U*S~*V'"
Matrix.show(R,document.body);
});
});


Hey - wait a minute .. what’s that string line inside the function ? And why is the function wrapped in a Matrix call ?

Lets look at another example to see how linalg.js will make your world easier. and more readable.

// Find the rigid transform that transforms pointset A to pointset B.
// A and B are matrices with nr_of_points rows and nr_of_dimensions columns.
// It returns a square nr_of_dimensions dimensional matrix T so that A*T=B
var findTransform = Matrix(function(A,B){
var mA, mB, U, S, V, COV;

// Center dataset by subtracting average row..
"mA = A-A.avg"
"mB = B-B.avg"

// Calculate covariance matrix.
"COV = mA'*mB"

// do SVD factorisation on covariance matrix.
"[U,S,V] = COV.SVD"

// Find transformation that brings A to B
"return U*V'"
});


That’s right .. full matrix math, factorisation and operator overloading .. just a string quote away from being javascript..

## enter linalg.js

12861 bytes - https://enkimute.github.io/res/linalg.min.js

All features of linalg.js are accessed through a single global function call, appropriately named ‘Matrix’.

#### you use it to create matrices ..

// a skinny matrix.
var A = Matrix([[1,2],[3,4],[5,6],[7,8]]);

// an empty matrix.
var B = Matrix(4,2);

// a matrix from an image, video or canvas :
var C = Matrix(document.getElementById('myImage'));


#### Or solve some simple expressions ..

// Solve (x=0,x=1,y=0)
var solution = Matrix("[[1,0],[1,0],[0,1]]\\[0,1,0]");

// add the transpose of a to b then multiply with c
var D = Matrix("(A'+B)*C");

// Multiply B with the inverse of A :
var E = Matrix("B*A^-1");

// you can store these expressions as functions.
// simpy add the arguments list as second parameter.
var solve = Matrix("(A'*A)^-1*A'*Y","A,Y");
var solution2 = solve([[1,0],[1,0],[0,1]],[0,1,0]);


#### or wrap your functions in it ..

.. so that you can put linear algebra expressions between string quotes on their own line ..

var sample = Matrix(function(){
var A=[[1,0],[1,0],[0,1]];
var Y=[0,1,0];
"return A\Y"
});

var solve = Matrix(function(A,Y){
var Ai = Matrix(A);
"Ai = (A'*A)^-1*A'"
"return Ai*Y"
});
solve([[1,0],[1,0],[0,1]],[0,1,0]);


#### Together with modern javascript features ..

you can write some pretty readable algebra code.

var fitCubic = Matrix(function (data) {
var Vandermonde = data.map(x=>[x[0]*x[0]*x[0],x[0]*x[0],x[0],1]);
var Rhs = data.map(x=>x[1]);
"return Vandermonde\Rhs"
});


In just a few lines we can implement various popular least squares methods ..

// Linear Least squares, regularised linear least squares, weighted linear least squares.
var lls  = Matrix("A\\Y","A,Y");
var rlls = Matrix("(A'*A + l*I)^-1*A'*Y","A,Y,I,l");
var wlls = Matrix("(A'*W~*A)^-1*A'*W~*Y","A,Y,W");

// Find intersection of the system ( y=0, x=1, y=2, x=3 )
var A=[[0,1],[1,0],[0,1],[1,0]];
var Y=[0,1,2,3];

// Using linear least squares
lls(A,Y).log();

// Regularised .. importance of arg size is 0.1
rlls(A,Y,Matrix.identity(2),0.1).log();

// Weighted .. first two more important.
wlls(A,Y,[1,1,0.5,0.5]).log();


## List of supported operators and functions.

All operators work with Matrix,Vector and Scalar types as appropriate. All functions will upcast their arguments to Matrix objects if needed.

Operator Description Example
~ Diagonalise vector into matrix. A=V~
() Parentheses D=(A+B)*C

Decomposition Expression
Matrix.factorCholesky(A); C=A.Cholesky
Matrix.factorLU(A); LU=A.LU
Matrix.factorLUP(A); [LU,P]=A.LUP
Matrix.factorSVD(A,S,V); [U,S,V]=A.SVD

Inversion Expression
Matrix.invertCholesky(A);
Matrix.invertLU(A);
Matrix.invertLUP(A); B=A^-1
Matrix.invertSVD(A);

Solver Expression
Matrix.solveCholesky(C,Y);
Matrix.solveLU(LU,Y);
Matrix.solveLUP(LU,P,Y);
Matrix.solveSVD(U,S,V,Y)
Matrix.solveSeidel(A,Y,iter);
Matrix.solve(A,Y); X=A\Y

Special Matrices Description
Matrix.identity(x); Identity matrix of size x
Matrix.hankel(V,c); Hankel matrix of column V and count c
Matrix.toHankel(A); Convert to Hankel form
Matrix.toeplitz(V,c); Toeplitz matrix of vector V and count c
Matrix.augment(A,B,AB); Augment A with B

Metrics/Norms Description
Matrix.equals(A,B); compare
Matrix.len(A); length matrix or vector
Matrix.normalize(A); matrix or vector
Matrix.avg(A); return average row.

Utility description
Matrix.cmul(A,B); component wise multiply
Matrix.print(A); pretty console log
Matrix.show(A,htmlElement) Append a matrix as image.
Matrix.row(A,x,c); extract rows from A
Matrix.repeat(r,x); repeat row r x times.

Filters Description
Matrix.blackman(n); Generate blackman vector of size n
Matrix.hamming(n); Generate hamming vector of size n
Matrix.sinc(n,sampling,cutoff); Generate a sinc vector of size n and given sampling and cutoff

Written on April 3, 2017