# Problem 13:
#
polyroot( c(4, -2, -4, 2) )

polyroot( c(4, 0, -5, 0, 1) )

polyroot( c(-1, 0, 3, 0, 3, 0, 1) )

polyroot( c(2, -3, 3, -3, 1) )

polyroot( c(4, 2, 1) )

polyroot( c(16, 0, 0, 0, 8, 0, 0, 0, 1) )

# Problem 10:
polyroot( c(1, 0, 2, 0, 1) )

polyroot( c(1, 3, -5, 1) )

polyroot( c(2, 6, 5, 1) )

# Problem 25:
polyroot( c(4, 14, 21, 18) ) 

# Problem 26:
polyroot( c(-36, 30, 6, -7, 1) )

# Problem 27:
polyroot( c(5, 37, 75, 31, 12) )

# Problem 28:
polyroot( c(14, 22, 17, 6, 1) )

# Problem 30:
A = matrix( c(1, 0, 1, 0,
              1, 1, -1, 1,
              0, 1, 0, -1,
              -1, 1, 1, 1), nrow=4, ncol=4, byrow=TRUE )
b = c(0, 0, -1, 0)
solve(A, b) 


# Problem 31:
A = matrix( c(1, 1, exp(2), exp(2),
              0, 1, 2 * exp(2), 3 * exp(2),
              0, 0, 4, 8 * exp(2),
              0, 0, 8 * exp(2), 20 * exp(2)), nrow=4, ncol=4, byrow=TRUE )
b = c(-1, 2, 0, 0)
solve(A, b) 


# Problem 32:
polyroot( c(-1, 1, -1, 1) )
A = matrix( c(1, 1, 0,
              1, 0, 1,
              1, -1, 0), 
           nrow=3, ncol=3, byrow=TRUE )
b = c(2, -1, -2)
print(solve(A, b))


# Problem 33:
polyroot( c(4, 4, -9, -1, 2) )
A = matrix( c(1, 1, 1, 1, 
              -1/2, 1, -2, 2, 
              1/4, 1, 4, 4,
              -1/8, 1, -8, 8), nrow=4, ncol=4, byrow=TRUE )
b = c(-2, 0, -2, 0)
print(solve(A, b))


# Problem 34:
polyroot( c(5, 1, 0, 4) )
A = matrix( c(1, 1, 0,
              -1, 1/2, 1,
              1, -3/4, 1), nrow=3, ncol=3, byrow=TRUE )
b = c(2, 1, -1)
print(solve(A, b))


# Problem 35:
polyroot( c(1, 5, 6) )
A = matrix( c(1, 1, 1,
              0, -1/3, -1/2,
              0, 1/9, 1/4), nrow=3, ncol=3, byrow=TRUE )
b = c(-2, 2, 0)
print(solve(A, b))


# Problem 36:
A = matrix( c(1, 0, 1, 0,
              -1, 1, -2, sqrt(3),
              0, -2, 1, -4 * sqrt(3),
              2, 2, 10, 9 * sqrt(3)), nrow=4, ncol=4, byrow=TRUE )
b = c(1, -2, 0, 3)
print(solve(A, b)) 


# Problem 39:
A = matrix( c(1, 0, 1, 0,
              0, 1, 0, sqrt(6), 
              -1, 0, -6, 0,
              0, -1, 0, -6*sqrt(6)),
           nrow=4, ncol=4, byrow=TRUE )
b = c(1, 0, -1, 0) # part (c)
b = c(-2, 0, 12, 0) # part (d)
print(solve(A, b))