# Project 5 for the University of Tulsa's CS-7863 Sci-Stat Course
# Systems of Equations and Finite Difference Methods
# Professor: Dr. McKinney, Spring 2023
# Noah L. Schrick - 1492657

## 1. Systems of Equations in Matrix Form
# a. Use built-in R solve
A <- matrix(c(2,3,4,3,2,-3,4,4,2),nrow=3)
b <- matrix(c(3,5,9), nrow=3)

x <- solve(A,b)
x

# b. Verify with matrix mult
all.equal(b, as.matrix(A %*% x))

## 2. Matrix from polynomial
# a. Create vectors for x and y
xvec <- c(-3,-1.5,0.5,2,5)
yvec <- c(6.8,15.2,14.5,-21.2,10)

# b. Create coefficient matrix
A.2 <- matrix(ncol=length(xvec), nrow=length(xvec))
for (i in 1:length(xvec)){
  A.2[,i] = xvec^(i-1)
}
A.2

# c. Solve 
if (!require("matlib")) install.packages("matlib")
library(matlib)

# verify
bvec <- solve(A.2, yvec)
all.equal(as.matrix(yvec), A.2 %*% bvec) # use all.equal instead of == (tols and storage type)
# EX: identical(as.double(8), as.integer(8)) returns FALSE



# d. Plot
poly_predict <- function(x){ 
  # given input x, returns polynomial prediction 
  sum(bvec*c(1,x,x^2,x^3,x^4))
}

xdomain <- seq(min(xvec),max(xvec),.2)    # predicted domain
y.predict <- sapply(xdomain,poly_predict) # predicted range
plot(xvec,yvec,ylim=c(-50,20))    # plot data
lines(xdomain,y.predict,type="l") # overlay solid line

## 3. Kirchoff
# a. Create vectors
# Resistance vec
# Voltage vec
# Matrix A and vector b
# Solve
# Show currents

# b. Repeat a. Use V=200, and R=(5,10,5,15,0,20)

## 4. Schrodinger eq
#a. Solve and plot the 1d quantum harmonic oscillator wave function

# b. Solve with Shooting Method

## 5. Solve harmonic oscillator Schrodinger with finite difference
# a. Solve with R and Julia

# b. Solve damped oscillator with perturbation param
# Plot

## 6. Solve heat diffusion equation

# EC. Solve the square plate problem