Loc(en) - Wrj4P5 Wiki - Wrj4P5

PackageLoc : Package of 3D-Vectors, general Vec/Mat/Functions

Outline

1. 1. PackageLoc : Package of 3D-Vectors, general Vec/Mat/Functions
      1. Loc : the model of 3D-Location and/or 3D-Vector
      2. Rod : the model of 3D-Line
      3. Tag : the model of 3D-Plane
      4. Ship : the model of 3D-Space
      5. Vec : the model of Linear Algebra with the linear operations.
      6. EqSys : the model of the General Equation System
      7. VFunc : the model of the Vector Function with params

this package is not only for Processing, but also for any situation, including,

• Loc : the model of 3D geometric point/location with geometric calculations.(Wrj4P5 needs Loc)
• Rod : the model of 3D geometric line with geometric calculations.
• Tag : the model of 3D geometric plane with geometric calculations.
• Ship : the model of 3D geometric space with geometric calculations. (intermediate)
• Vec : the model of Linear Algebra with the linear operations, having root finder.
• Mat : the model of Linear Algebra with the linear operation, having some linear solvers.
• EqSys : the model of the General Equation System, having some solvers. (abstract class)
• Vfunc : the mdel of the Vector Function with some parameters, having root/extreme finder and parameter estimator. (abstract class)

the test codes(sketch) for Vec,Mat, EqSys, Vfunc is here

the test codes(sketch) for Rod is here

Loc : the model of 3D-Location and/or 3D-Vector

The model of geometric point with 3D location and geometric calculations.

Rod : the model of 3D-Line

having the following image

• at(t)
• stern()
• fore()
• bow()
• length()

Tag : the model of 3D-Plane

adding the width to Rod

• at(t,s)
• keel()
• starBoard()
• star()
• width()
• normal()

Ship : the model of 3D-Space

adding the height to Tag

• at(t,s,u)
• sea()
• bridgeTop()
• up()
• height()

Vec : the model of Linear Algebra with the linear operations.

The index of Vector is 0-based -- e.g.,

elem(0) : the left most element
elem(idx) : the (idx+1)th. element.

can solve the polynomial equation, all the roots are complex with DKA method

c₀*xⁿ + c₁*x^(n-1) + .... + c_n-1*x¹ = b

you can use the convenient, Vec realRoots(Mat root) to get unique real roots if exist.

Mat : the model of Linear Algebra with the linear operations

the model of Linear Algebra with the linear operations.
The index of Matrices is 0-based -- e.g.,

elem(0, 0) : the element in the first row, first column
elem(n, m) : the element in the (n+1)th. row, (m+1)th column.

can solve the linear equations, with LU decomposition

{ {c_ij} } * { x_j } = { b_i ]

[Caution!]

The LU matrix is cached and reused on subsequent calls,

solve(), isSingular(), det(), inverse()

If data are modified via references getDataRef(), then the stored LU decomposition will not be discarded. In this case, you need to explicitly invoke LUDecompose() to recompute The LU decomposition is performed before using any of the methods above.

EqSys : the model of the General Equation System

Model of multi-value Vector Functions, the Equation System

f(x) = [ f_i(x) | f_i(x) : Rⁿ -> R, i=0->(m-1) ]

can solve the multi-value non-linear equation, with Simplex(Nelder) and Newton

f(x) = [bi]

VFunc : the model of the Vector Function with params

Model of the Vector Function with some parameters

f(x:p) : Rⁿ,R^m -> R

can solve the followings with Simplex(Nelder) and Newton

• for given p, find x such that f(x;p) = 0
• for given p, find extreme point x* such that f(x*;p)/dx = 0
• estimate parameters p* such that ssr(obs,samples,p*)/dp = 0, based the set of the observations, {obs_i in R, samples_i in Rⁿ}