Assignment 1

Fundamentals of Computational Mathematics

CUNY MSDS DATA 605, Fall 2018

Rose Koh

08/30/2018

u <- c(0.5, 0.5)
v <- c(3, -4)
library(pracma)
uv.dot <- dot(u,v) #or u %*% v
uv.dot

## [1] -0.5

u.len <- sqrt(sum(u^2))
u.len

## [1] 0.7071068

v.len <- sqrt(sum(v^2))
v.len

## [1] 5

3*u - 2*v

## [1] -4.5  9.5

rad <- acos(uv.dot / (u.len * v.len))
# in degrees
rad * 180/pi

## [1] 98.1301

A <- matrix(c(1,1,3,2,-1,5,-1,-2,4), 3, byrow=T)
b <- matrix(c(1,2,6), nrow=3, ncol=1)

# Elimination Function
elimination <- function(A,b){
  
  aug.mat <- cbind(A,b)         # bind A,b to create an Augmented matrix
                                # devide the first row by the coef of the first row and first column
  aug.mat[1,] <- aug.mat[1,]/aug.mat[1,1] 
  
  print("the coef [1,1] should be 1.")
  print(aug.mat)                            
  
  p <- nrow(A)
  
  i <- 2                        # start with a second row
  while (i < p+1) {             # iterate through all rows in augmented matrix
   j <- i                       # assign row num to col
   while (j < p+1) {            # iterate through all columns in augmented matrix
     
                                # multiply the prev row by coef of current row,col 
                                # that equals the num of prev row, 
                                # then subtract it from the current row
    aug.mat[j, ] <- aug.mat[j, ] - aug.mat[i-1, ] * aug.mat[j, i-1]
    j <- j+1                    # iterate next column
   }
                                # bind augmented matrix that contains current row only 
                                # with augmented matrix that doesn't contain current row 
                                # while the coef in current row, current row equals 0
   while (aug.mat[i,i] == 0) { 
    aug.mat <- rbind(aug.mat[-i,],aug.mat[i,])
   }
   aug.mat[i,] <- aug.mat[i,]/aug.mat[i,i]
   i <- i+1                     # iterate next row
  }
  
  print("the coefficients located in [1,1],[2,2],[3,3] etc. (where row=column) should equal to 1")
  print(aug.mat)
  
 
  for (i in p:2){               # iterate through rows from last row to the second row
   for (j in i:2-1) {           # iterate through columns (from the first)
                                # multiply the current row by coef located in the same row and current col
                                # then subtract it from the row that equals to the current col
     aug.mat[j, ] <- aug.mat[j, ] - aug.mat[i, ] * aug.mat[j, i]
   }
  }
  print("the coefficients located in [1,1],[2,2],[3,3] etc. (where row=column) should equal to 1 while all other coefficients shoud equal to 0")
  print(aug.mat)
}

Solve

elimination(A,b)

## [1] "the coef [1,1] should be 1."
##      [,1] [,2] [,3] [,4]
## [1,]    1    1    3    1
## [2,]    2   -1    5    2
## [3,]   -1   -2    4    6
## [1] "the coefficients located in [1,1],[2,2],[3,3] etc. (where row=column) should equal to 1"
##      [,1] [,2]      [,3]      [,4]
## [1,]    1    1 3.0000000 1.0000000
## [2,]    0    1 0.3333333 0.0000000
## [3,]    0    0 1.0000000 0.9545455
## [1] "the coefficients located in [1,1],[2,2],[3,3] etc. (where row=column) should equal to 1 while all other coefficients shoud equal to 0"
##      [,1] [,2] [,3]       [,4]
## [1,]    1    0    0 -1.5454545
## [2,]    0    1    0 -0.3181818
## [3,]    0    0    1  0.9545455

Double check

# compare the result with solve function
solve(A,b)

##            [,1]
## [1,] -1.5454545
## [2,] -0.3181818
## [3,]  0.9545455