LU decomposition

Description

The function decomposes matrix A into LU with L lower matrix and U as upper matrix

Usage

LU.decompose(A, tol = 1e-07)

Arguments

Value

A : Transformed matrix with Upper part of the triangele is (U) and Lower part of the triangular is L (Diagnoal for the L matrix is 1)

Examples

A<-matrix(c(0,-1,1, -1,2,-1,2,-1,0), nrow=3,ncol=3, byrow = TRUE)
Z<-optR(A, tol=1e-7, method="LU")

Solving system of equations using LU decomposition

Description

The function solves Ax=b using LU decomposition (LUx=b). The function handles multple responses

Usage

LU.optR(A, b, tol = 1e-07)

Arguments

Value

U : Upper part of the triangele is (U) and Lower part of the triangular is L (Diagnoal for the L matrix is 1)

c : Lc=b (where Ux=c)

beta : Estimates

Examples

A<-matrix(c(0,-1,1, -1,2,-1,2,-1,0), nrow=3,ncol=3, byrow = TRUE)
b<-matrix(c(0,0, 1), nrow=3,ncol=1,byrow=TRUE)
Z<-optR(A, b, method="LU")

Function to extract Lower and Upper matrix from LU decomposition

Description

function to extract Lower and Upper matrix from LU decomposition

Usage

LUsplit(A)

Arguments

Value

U : upper triangular matrix

L : Lower triangular matrix

Examples

A<-matrix(c(0,-1,1, -1,2,-1,2,-1,0), nrow=3,ncol=3, byrow = TRUE) 
Z<-optR(A, method="LU")
LUsplit(Z$U)

Optimization & estimation based on Conjugate Gradient Method

Description

Function utilizes the Conjugate Gradient Method for optimization to solve equation Ax=b

Usage

cgm(A, b, x = NULL, iter = 500, tol = 1e-07)

Arguments

Value

optimal : Optimal solutions

initial : initial solution

itr.conv : Number of iterations for convergence

conv : Convergence array

Examples

A<-matrix(c(4,-1,1, -1,4,-2,1,-2,4), nrow=3,ncol=3, byrow = TRUE)
b<-matrix(c(12,-1, 5), nrow=3,ncol=1,byrow=TRUE)
Z<-optR(A, b, method="cgm", iter=500, tol=1e-7)

Function for Choleski Decomposition

Description

Function perform choleski decomposition for positive definate matrix (A=LL')

Usage

choleskiDecomposition(A, tol = 1e-07)

Arguments

Value

L: Decomposition matrix

Examples

A<-matrix(c(4,-2,2, -2,2,-4,2,-4,11), nrow=3,ncol=3, byrow = TRUE)
chdA<-choleskiDecomposition(A)

Function fits linear model using Choleski Decomposition

Description

Function fits a linear model using Choleski Decomposition for positive definate matrix

Usage

choleskilm(A, b, tol = 1e-07)

Arguments

Value

U : Upper part of the triangele is (U) and Lower part of the triangular is L (Diagnoal for the L matrix is 1)

c : Lc=b (where Ux=c)

beta : Estimates

examples A<-matrix(c(6,-4,1, -4,6,-4,1,-4,6), nrow=3,ncol=3, byrow = TRUE) b<-matrix(c(-14,36, 6), nrow=3,ncol=1,byrow=TRUE) Z<-optR(A, b, method="choleski") # Solve Linear model using Gauss Elimination

Function to solve linear system using forward substitution

Description

Function to solve linear system using backsubsitution using Upper Triangular Matrix (Ux=c)

Usage

forwardsubsitution.optR(L, b)

Arguments

Value

y : Estiamted value

Gauss-Seidel based Optimization & estimation

Description

Function utilizes the Gauss-Seidel optimization to solve equation Ax=b

Usage

gaussSeidel(A, b, x = NULL, iter = 500, tol = 1e-07, w = 1,
  witr = NULL)

Arguments

Value

optimal : Optimal solutions

initial : initial solution

relaxationFactor : relaxation factor

Examples

A<-matrix(c(4,-1,1, -1,4,-2,1,-2,4), nrow=3,ncol=3, byrow = TRUE)
b<-matrix(c(12,-1, 5), nrow=3,ncol=1,byrow=TRUE)
Z<-optR(A, b, method="gaussseidel", iter=500, tol=1e-7)

Function determines the Hat matrix or projection matrix for given X

Description

Function hatMatrix determines the projection matrix for X from the form yhat=Hy. The projection matrix defines the influce of each variable on fitted value The diagonal elements of the projection matrix are the leverages or influence each sample has on the fitted value for that same observation. The projection matrix is evaluated with I.I.D assumtion ~N(0, 1)

Usage

hatMatrix(X, covmat = NULL)

Arguments

Value

X: Projection Matrix

Examples

X<-matrix(c(6,-4,1, -4,6,-4,1,-4,6), nrow=3,ncol=3, byrow = TRUE)
covmat <- matrix(rnorm(3 * 3), 3, 3)
H<-hatMatrix(X)
H<-hatMatrix(X, covmat)
diag(H)

Invert a matrix using LU decomposition

Description

function invert a matrix A using LU decomposition such that A*inv(A)=I

Usage

inv.optR(A, tol = 1e-07)

Arguments

Value

A : Inverse of Matrix A

Examples

# Invert the matrix using LU decomposition
A<-matrix(c(0.6,-0.4,1, -0.3,0.2,0.5,0.6,-1,0.5), nrow=3,ncol=3, byrow = TRUE) 
InvA<-inv.optR(A)

Function to evaluate jacobian matrix from functions

Description

jacobian is function to determine the jacobian matrix for function f for input x

Usage

jacobian(f, x)

Arguments

Value

jacobiabMatrix: Jacobian matrix

f0: Intial solution

Function to address machine precision error

Description

function to remove the machine precision error

Usage

machinePrecision(A)

Arguments

Value

A : return matrix

Function for Newton Rapson roots for given equations

Description

newtonRapson function perform optimization

Usage

newtonRapson(f, x, iteration = 30, tol = 1e-09)

Arguments

Value

x : optimal roots

Non-diagnoal multipication

Description

Function for non-diagnoal multipication

Usage

nonDiagMultipication(i, A, beta)

Arguments

Value

asum: Non-diagnol contribution

Function to Re-order the matrix to make dominant diagnals

Description

Function re-order the matrix to make matrix pivot.diag at each iteration

Usage

opt.matrix.reorder(A, tol = 1e-16)

Arguments

Value

A : Updated Matrix

b.order : Order sequence of A updated matrix

Examples

A<-matrix(c(0,-1,1, -1,2,-1,2,-1,0), nrow=3,ncol=3, byrow = TRUE)
b<-matrix(c(0,0, 1), nrow=3,ncol=1,byrow=TRUE)
Z<-optR(A, b, method="gauss")

Optimization & predictive modelling Toolsets

Description

optR function for solving linear systems using numerical approaches. Current toolbox supports Gauss Elimination, LU decomposition, Conjugate Gradiant Decent and Gauss-Sideal methods for solving the system of form AX=b For optimization using numerical methods cgm method performed faster in comparision with gaussseidel. For decomposition LU is utilized for multiple responses to enhance the speed of computation.

Usage

optR(x, ...)

Arguments

Value

optR : Return optR class

Author(s)

PKS Prakash

Examples

# Solving equation Ax=b
A<-matrix(c(6,-4,1, -4,6,-4,1,-4,6), nrow=3,ncol=3, byrow = TRUE)
b<-matrix(c(-14,36, 6), nrow=3,ncol=1,byrow=TRUE)
Z<-optR(A, b, method="gauss") # Solve Linear model using Gauss Elimination

# Solve Linear model using LU decomposition (Supports Multi-response)
Z<-optR(A, b, method="LU") 

# Solve the matrix using Gauss Elimination (1, -1, 2)
A<-matrix(c(2,-2,6, -2,4,3,-1,8,4), nrow=3,ncol=3, byrow = TRUE)
b<-matrix(c(16,0, -1), nrow=3,ncol=1,byrow=TRUE)
Z<-optR(A, b, method="gauss") # Solve Linear model using Gauss Elimination

require(utils)
set.seed(129)
n <- 10 ; p <- 4
X <- matrix(rnorm(n * p), n, p) # no intercept!
y <- rnorm(n)
Z<-optR(X, y, method="cgm")

Function to solve linear system using backsubsitution

Description

Function to solve linear system using backsubsitution using Upper Triangular Matrix (Ux=c)

Usage

## S3 method for class 'backsubsitution'
optR(U, c)

Arguments

Value

beta : Estiamted value

Optimization & predictive modelling Toolsets

Description

soptR is the default function for optimization

Usage

## Default S3 method:
optR(x, y = NULL, weights = NULL, method = c("gauss",
  "LU", "gaussseidel", "cgm", "choleski"), iter = 500, tol = 1e-07,
  keep.data = TRUE, ...)

Arguments

Value

U : Decomposed matrix for Gauss-ELimination Ax=b is converted into Ux=c where U is upper triangular matrix for LU decomposition U contain the values for L & U decomposition LUx=b

c : transformed b & for LU transformation c is y from equation Ux=y

estimates : Return x values for linear system

seq : sequence of A matrix re-ordered

Examples

# Solving equation Ax=b
A<-matrix(c(6,-4,1, -4,6,-4,1,-4,6), nrow=3,ncol=3, byrow = TRUE)
b<-matrix(c(-14,36, 6), nrow=3,ncol=1,byrow=TRUE)
Z<-optR(A, b, method="gauss") 

# Solve Linear model using LU decomposition (Supports Multi-response)
Z<-optR(A, b, method="LU")

# Solving the function using numerical method
Z<-optR(A, b, method="cgm")

require(utils)
set.seed(129)
n <- 7 ; p <- 2
X <- matrix(rnorm(n * p), n, p) # no intercept!
y <- rnorm(n)
Z<-optR(X, y, method="LU")

Fitter function for Linear/Non-linear system with form Ax=b

Description

optR.fit is fit function for determing x for System with form Ax=b

Usage

## S3 method for class 'fit'
optR(x, y = NULL, method = c("gauss, LU, gaussseidel", "cgm"),
  iter = 500, tol = 1e-07, ...)

Arguments

Value

U : Decomposed matrix for Gauss-ELimination Ax=b is converted into Ux=c where U is upper triangular matrix for LU decomposition U contain the values for L & U decomposition LUx=b

c : transformed b & for LU transformation c is y from equation Ux=y

estimates : Return x values for linear system

seq : sequence of A matrix re-ordered

Examples

# Solving equation Ax=b
A<-matrix(c(6,-4,1, -4,6,-4,1,-4,6), nrow=3,ncol=3, byrow = TRUE)
b<-matrix(c(-14,36, 6), nrow=3,ncol=1,byrow=TRUE)
Z<-optR(A, b, method="gauss") 

# Solve Linear model using LU decomposition (Supports Multi-response)
Z<-optR(A, b, method="LU")

# Solving the function using numerical method
Z<-optR(A, b, method="cgm")

Optimization & predictive modelling Toolsets

Description

optR package to perform the optimization using numerical methods

Usage

## S3 method for class 'formula'
optR(formula, data = list(), weights = NULL,
  method = c("gauss, LU, gaussseidel", "cgm", "choleski"), iter = 500,
  tol = 1e-07, keep.data = TRUE, contrasts = NULL, ...)

Arguments

Value

U : Decomposed matrix for Gauss-ELimination Ax=b is converted into Ux=c where U is upper triangular matrix for LU decomposition U contain the values for L & U decomposition LUx=b

c : transformed b & for LU transformation c is y from equation Ux=y

estimates : Return x values for linear system

Author(s)

PKS Prakash

Examples

# Solving equation Ax=b
b<-matrix(c(-14,36, 6), nrow=3,ncol=1,byrow=TRUE)
A<-matrix(c(6,-4,1, -4,6,-4,1,-4,6), nrow=3,ncol=3, byrow = TRUE)
Z<-optR(b~A-1, method="gauss") # -1 to remove the constant vector

Z<-optR(b~A-1, method="LU") # -1 to remove the constant vector

require(utils)
set.seed(129)
n <- 10 ; p <- 4
X <- matrix(rnorm(n * p), n, p) # no intercept!
y <- rnorm(n)
data<-cbind(X, y)
colnames(data)<-c("var1", "var2", "var3", "var4", "y")
Z<-optR(y~var1+var2+var3+var4+var1*var2-1, data=data.frame(data), method="cgm")

gauss to solve linear systems

Description

Function solves linear systems using Gauss Elimination. The function solves equation of form Ax=b to Ux=c (where U is upper triangular matrix)

Usage

## S3 method for class 'gauss'
optR(A, b, tol = 1e-07)

Arguments

Value

U : Upper triangular matrix

c : Transformed b

beta : Estimates

Examples

A<-matrix(c(0,-1,1, -1,2,-1,2,-1,0), nrow=3,ncol=3, byrow = TRUE)
b<-matrix(c(0,0, 1), nrow=3,ncol=1,byrow=TRUE)
Z<-optR(A, b, method="gauss")

Function to estimate lambda

Description

Function esimates the lambda or multiplier factor for Elimination using the pivot row/column

Usage

## S3 method for class 'multiplyfactor'
optR(rowindex, A, pivotindex)

Arguments

Value

lambda : Lambda

Function based optimization module

Description

Function based optimization module

Usage

optRFun(formula, x0, iteration = 30, method = c("newtonrapson"),
  tol = 1e-09)

Arguments

Value

optRFun : Optimal Solution class

Function based optimization module

Description

Newton Rapshon based optimization

Usage

optRFun.newtonRapson(formula, x0, iteration = 30, tol = 1e-09)

Arguments

Value

optRFun : Optimal Solution

Prediction function based on optR class

Description

Function for making predictions using OptR class

Usage

## S3 method for class 'optR'
predict(object, newdata, na.action = na.pass, ...)

Arguments

Value

fitted.val : Predicted values

terms : terms used for fitting

print coefficients for optR class

Description

optR is the default function for optimization

Usage

## S3 method for class 'optR'
print(x, ...)

Arguments

Examples

# Solving equation Ax=b
A<-matrix(c(6,-4,1, -4,6,-4,1,-4,6), nrow=3,ncol=3, byrow = TRUE)
b<-matrix(c(-14,36, 6), nrow=3,ncol=1,byrow=TRUE)
Z<-optR(A, b, method="gauss")
print(Z)

Generate Summary for optR class

Description

summary function generates the summary for the optR class

Usage

## S3 method for class 'optR'
summary(object, ...)

Arguments

Examples

# Solving equation Ax=b
A<-matrix(c(6,-4,1, -4,6,-4,1,-4,6), nrow=3,ncol=3, byrow = TRUE)
b<-matrix(c(-14,36, 6), nrow=3,ncol=1,byrow=TRUE)
Z<-optR(A, b, method="cgm")
summary(Z)