605HW1

Problem 01

(1) Calculate the dot product u:v where u = [0.5;0.5] and v= [3;-4]

u <- c(0.5,0.5)
v <- c(3,-4)

##Dot product without using R function
dotprod<-u[1]*v[1]+u[2]*v[2]
dotprod

## [1] -0.5

##Dot product using R function
library('pracma')
dotprod1<-dot(u,v)
dotprod1

## [1] -0.5

(2) What are the lengths of u and v?

##Length u
len_u<-sqrt(u[1]^2+u[2]^2)
len_v<-sqrt(v[1]^2+v[2]^2)
len_u

## [1] 0.7071068

len_v

## [1] 5

(3) What is the linear combination 3u-2v?

lin_comb<-3*u-2*v
lin_comb

## [1] -4.5  9.5

(4) What is the angle between u and v?

##The angle is Cosinv(dotprod/len_u*len_v)
angle_rad<-acos(dotprod/(len_u*len_v))
angle_degree<-angle_rad*180/pi
##Angle in degrees is 98.13
angle_degree

## [1] 98.1301

Problem 02

Write a function in R that takes matrix A and constraint vector b and solve using upper triangle and backtesting

matsolv<-function(A,b){
  
  cmat<-cbind(A,b)##Bind A and b
  
  ## Check for 0 in top left pivot
  if(cmat[1,1]==0){
    ##Check if cmat1[2,2]is non zero
    if(cmat[2,1]!=0){
      cmat = cmat[c(2,1,3),]
    } ##Switch R1 with R2
    else{
      cmat=cmat(c(3,2,1)) ##Switch R1 with R3
    }
  }

  ## Check if row1 col1 has 1, else divide by first element to get 1
  if (cmat[1,1] != 1) {
    cmat[1,] <- cmat[1,] / cmat[1,1]
  }

  ## Convert item [2,1] to 0
  if (cmat[2,1] !=0) {
    # Multiply it by each item in R1 and then adjust to make [2,1] to 0
    adjustment <- cmat[2,1] * cmat[1,]
    cmat[2,] <- cmat[2,] - adjustment
  }

  ## Convert item [3,1] to 0
  if (cmat[3,1] !=0) {
    # use value of row 3 column 1 and multiply by 1, then deduct to make [3,1] to 0
    adjustment <- cmat[3,1] * cmat[1,]
    cmat[3,] <- cmat[3,] - adjustment
  }

  ## Check if row2col2 is 0, if yes, switch with row 3 to get non zero, divide by [2,2] to get 1
  if (cmat[2,2] == 0) {
    cmat = cmat[c(1,3,2),]
  } 
  cmat[2,] <- cmat[2,] / cmat[2,2]

  ## Convert [3,2] to 0
  adjustment <- cmat[3,2] * cmat[2,]
  cmat[3,] <- cmat[3,] - adjustment

  ##Solve with back substitution
  x3<- cmat[3,4]/cmat[3,3]
  x2<- -cmat[2,3]
  x1<- 1-x2-3*x3

  ##Return the result vector
  result <- as.vector(round(c(x1,x2,x3), digits=2))
  return(result)
}

##Testing with R function
A <- matrix(c(1,1,3,2,-1,5,-1,-2,4), ncol=3, byrow=TRUE)
b <- c(1,2,6)
matsolv(A,b)

## [1] -1.53 -0.33  0.95

Write a function in R that takes matrix A and constraint vector b and solve using Gauss Jordan elimination

matsolv1<-function(A,b){
  
  mat1<-cbind(A,b)##Bind the two to make augmented matrix
  
  ##Transform R2=-2R1+R2
  mat1[2,]<- -2*mat1[1,]+mat1[2,]
  
  ##Transform R3=R1+R3
  mat1[3,]<- mat1[1,]+mat1[3,]
  
  ##Transform R1=R1+R3
  mat1[1,]<- mat1[1,]+mat1[3,]
  
  ##Transform R2=-3R3+R2
  mat1[2,]<- -3*mat1[3,]+mat1[2,]
  
  ##Substitution
  x3<- mat1[2,4]/mat1[2,3]
  x1<- mat1[1,4]-10*x3
  x2<- -1*(7-7*x3)

  
  ##Return the result vector
  result <- as.vector(round(c(x1,x2,x3), digits=2))
  
  return(result)
}

##Testing with R function
A <- matrix(c(1,1,3,2,-1,5,-1,-2,4), ncol=3, byrow=TRUE)
b <- c(1,2,6)
matsolv1(A,b)

## [1] -1.55 -0.32  0.95