** DATA_605_Assignment_1_Thonn **
** Problem Set 1 **
u <- c(0.5, 0.5)
v <- c(3, -4)
dotprod1 <- u %*% v
dotprod1## [,1]
## [1,] -0.5lengthu <- sqrt(u %*% u)
lengthu## [,1]
## [1,] 0.7071068lengthv <- sqrt(v %*% v)
lengthv## [,1]
## [1,] 5lincombination1 <- 3*u - 2*v
lincombination1## [1] -4.5 9.5angle_rad <- acos((u %*% v)/(sqrt(u %*% u)*sqrt(v %*% v)))
angle_rad## [,1]
## [1,] 1.712693** Problem Set 2 **
Set up a system of equations with 3 variables and 3 constraints and solve for x. Please write a function in R that will take two variables (matrix A & constraint vector b) and solve using elimination. Your function should produce the right answer for the system of equations for any 3-variable, 3-equation system. You don’t have to worry about degenerate cases and can safely assume that the function will only be tested with a system of equations that has a solution. Please note that you do have to worry about zero pivots, though. Please note that you should not use the built-in function solve to solve this system or use matrix inverses. The approach that you should employ is to construct an Upper Triangular Matrix and then back-substitute to get the solution. Alternatively, you can augment the matrix A with vector b and jointly apply the Gauss Jordan elimination procedure.
Please test it with the system below and is should produce a solution x = [-1.55,-0.32,0.95]
A = matrix(c(1, 2, -1, 1, -1, -2, 3, 5, 4), nrow=3, ncol=3) b = c(1, 2, 6)
# Function 1 - swap elements - vector
elements_swap <- function(v, m, n){
a_v <- v[m]
v[m] <- v[n]
v[n] <- a_v
return(v)
}
# Function 2 - swap rows - matrix
rows_swap <-function(A, m, n){
a <- A[m,]
A[m,] <- A[n,]
A[n,] <- a
return(A)
}
# Function 3 - find pivot
pivot_find <- function (A, r, c){
# Initialize variables
swap_a <- FALSE
pv <- 0
swap_r <- 0
loop1 <- TRUE
j <- r
#loop thru each matrix and locate non-zero pivot point for column c
while(loop1){
if(A[j,c] != 0){
loop1 <- FALSE
pv<-A[j,c]
if(j>r){
swap_r <- j
swap_a <- TRUE
}
}
j <- j + 1
if (j > nrow(A)){
loop <- FALSE
}
}
return(list(pv, swap_a, swap_r))
}
# Main function to Solve equation (3x3)
solve_main <- function (A, b){
# Check input parameters: A, b, and whether row(A) equals length(b)
if(missing(A)){
stop("matrix A not in form Ax = b")
}
if(missing(b)){
stop("vector b not in form Ax = b")
}
if(nrow(A) != ncol(A)){
stop("A not a square matrix")
}
if(nrow(A) != length(b)){
stop("number of rows of A != dim of b")
}
num_rows <- nrow(A)
num_col <- ncol(A)
A1 <- A
b1 <- b
# Create Upper Triangular Matrix
# Initialize column cursor
c <- 1
while (c <= num_col){
# Initialize row cursor
r <- c
# find pivot point
l_result <- pivot_find (A1, r, c)
p <- unlist(l_result[1])
swap_indicator <- unlist(l_result[2])
swap_row <- unlist(l_result[3])
# Validate that pivot point exists
if (p==0){
text <- "No pivot point found in column"
txt_col <- as.character(c)
msg <- paste(text,txt_col)
stop(msg)
}
# if pivot point check for swap
if(swap_indicator){
A1 <- rows_swap(A1,r,swap_row)
b1 <- elements_swap (b1, r, swap_row)
}
# Process elimination for each row
k <- r + 1
while (k <= num_rows){
# multiplier check
m = A1[k,c]/p
A1[k, ] <- A1[k, ] - m*A1[r, ]
b1[k] <- b1[k] - m*b1[r]
k <- k + 1
} # end loop rows
c <- c + 1
} # end loop columns
# Solve equations
equation_v <- vector()
i <- num_col
equation_v[i] <- b1[i]/A1[i,i]
z<- b1[3]/A1[3,3]
y<- (b1[2] - A1[2,3]*z)/A1[2,2]
x<- (b1[1] - A1[1,2]*y - A1[1,3]*z)/A1[1,1]
result1 <- c(round(x, digits = 2), round(y, digits = 2), round(z, digits = 2))
return(result1)
}
# Test the Main Function #
A = matrix(c(1, 2, -1, 1, -1, -2, 3, 5, 4), nrow=3, ncol=3)
b = c(1, 2, 6)
result <- solve_main(A,b)
# actual result:
result## [1] -1.55 -0.32 0.95# expected result: [1] -1.55 -0.32 0.95END