eigen-1.1.1: Eigen C++ library (linear algebra: matrices, vectors, numerical solvers, and related algorithms).

Data.Eigen.LA

Description

The problem: You have a system of equations, that you have written as a single matrix equation

`Ax = b`

Where A and b are matrices (b could be a vector, as a special case). You want to find a solution x.

The solution: You can choose between various decompositions, depending on what your matrix A looks like, and depending on whether you favor speed or accuracy. However, let's start with an example that works in all cases, and is a good compromise:

```import Data.Eigen.Matrix
import Data.Eigen.LA

main = do
let
a = fromList [[1,2,3], [4,5,6], [7,8,10]]
b = fromList [,,]
x = solve ColPivHouseholderQR a b
putStrLn "Here is the matrix A:" >> print a
putStrLn "Here is the vector b:" >> print b
putStrLn "The solution is:" >> print x
```

produces the following output

```Here is the matrix A:
Matrix 3x3
1.0 2.0 3.0
4.0 5.0 6.0
7.0 8.0 10.0

Here is the vector b:
Matrix 3x1
3.0
3.0
4.0

The solution is:
Matrix 3x1
-2.0000000000000004
1.0000000000000018
0.9999999999999989
```

Checking if a solution really exists: Only you know what error margin you want to allow for a solution to be considered valid.

You can compute relative error using `norm (ax - b) / norm b` formula or use `relativeError` function which provides the same calculation implemented slightly more efficient.

Synopsis

# Basic linear solving

```Decomposition           Requirements on the matrix          Speed   Accuracy

PartialPivLU            Invertible                          ++      +
FullPivLU               None                                -       +++
HouseholderQR           None                                ++      +
ColPivHouseholderQR     None                                +       ++
FullPivHouseholderQR    None                                -       +++
LLT                     Positive definite                   +++     +
LDLT                    Positive or negative semidefinite   +++     ++
JacobiSVD               None                                -       +++

The best way to do least squares solving for square matrices is with a SVD decomposition (JacobiSVD)
```

Constructors

 PartialPivLU LU decomposition of a matrix with partial pivoting. FullPivLU LU decomposition of a matrix with complete pivoting. HouseholderQR Householder QR decomposition of a matrix. ColPivHouseholderQR Householder rank-revealing QR decomposition of a matrix with column-pivoting. FullPivHouseholderQR Householder rank-revealing QR decomposition of a matrix with full pivoting. LLT Standard Cholesky decomposition (LL^T) of a matrix. LDLT Robust Cholesky decomposition of a matrix with pivoting. JacobiSVD Two-sided Jacobi SVD decomposition of a rectangular matrix.

Instances

 Enum Decomposition Show Decomposition
x = solve d a b
finds a solution `x` of `ax = b` equation using decomposition `d`
e = relativeError x a b
computes `norm (ax - b) / norm b` where `norm` is L2 norm

# Multiple linear regression

linearRegression :: [[Double]] -> ([Double], Double) Source

(coeffs, error) = linearRegression points
computes multiple linear regression `y = a1 x1 + a2 x2 + ... + an xn + b` using `ColPivHouseholderQR` decomposition
• point format is `[y, x1..xn]`
• coeffs format is `[b, a1..an]`
• error is calculated using `relativeError`
```import Data.Eigen.LA
main = print \$ linearRegression [
[-4.32, 3.02, 6.89],
[-3.79, 2.01, 5.39],
[-4.01, 2.41, 6.01],
[-3.86, 2.09, 5.55],
[-4.10, 2.58, 6.32]]
```

produces the following output

```([-2.3466569233817127,-0.2534897541434826,-0.1749653335680988],1.8905965120153139e-3)
```