def P13():
print("---------------------------------")
print("P13: Page 127 Problem 7")
Q13_X = np.array([-3, 2, -1, 3, 1], dtype=float)
Q13_Y = np.array([0, 5, -4, 12, 0], dtype=float)
np2latex(np.vstack((Q13_X, Q13_Y)), "Top:X,Bot:Y")
Q13_coef = coeffts(Q13_X, Q13_Y)
np2latex(transpose(Q13_coef), "\\text{Newton Coef}")
def P11():
print("---------------------------------")
print("P11: Page 126 Problem 1")
Q11_X = np.array([-1.2, 0.3, 1.1])
Q11_Y = np.array([-5.76, -5.61, -3.69])
np2latex(np.vstack((Q11_X, Q11_Y)), "Top:X,Bot:Y")
print(" \\item Neville's Method")
Q11_X0_neville = neville(Q11_X, Q11_Y, 0)
print(" At x = 0: y = "+str(round(Q11_X0_neville, 5)))
print(" \\item Lagrange's Method")
Q11_X0_lagrange = lagrange(Q11_X, Q11_Y, 0)
print(" At x = 0: y = "+str(round(Q11_X0_lagrange, 5)))
def P12():
print("---------------------------------")
print("P12: Page 126 Problem 6")
#The program should return "divided difference table"
# and "degree of the polynomial" respectively.
#Hint: Look at page 113 of the book to learn how to
# find the degree of a polynomial from the
# divided difference table.
Q12_X = np.array([-2, 1, 4, -1, 3, -4], dtype=float)
Q12_Y = np.array([-1, 2, 59, 4, 24, -53], dtype=float)
np2latex(np.vstack((Q12_X, Q12_Y)), "Top:X,Bot:Y")
n=len(Q12_X)
#make divided difference table
dd = np.zeros([n,n])
dd[:,0] = Q12_Y
for i in range(1,n):
for o in range(1,i+1):
#can recursively find divided difference from previous column
dd[i,o] = (dd[i,o-1]-dd[o-1,o-1])/(Q12_X[i]-Q12_X[o-1])
np2latex(dd, "\\text{Divided Diff}")
#last nonzero is the order
print("Polynomial is of order " + str(max([i for i, arg in enumerate(dd.diagonal()) if arg != 0])))
def P7():
print("P7: Page 78 Problem 5")
Q7_A = np.array([[4, -2, 1],
[-2, 1, -1],
[-2, 3, 6]], dtype=float)
Q7_B = np.array([2, -1, 0], dtype=float)
np2latex(Q7_A, "A")
np2latex(transpose(Q7_B), "B")
Q7_X = gausPivot(Q7_A, Q7_B)
np2latex(transpose(Q7_X), "X")
def P10():
print("---------------------------------")
print("P10: Page 82 Problem 19")
for n in [2, 3, 4]:
Q10_A = genQ19_A(n)
Q10_B = genQ19_B(Q10_A)
np2latex(Q10_A, "A_{"+str(n)+"}")
np2latex(transpose(Q10_B), "B_{"+str(n)+"}")
Q10_X = gausPivot(Q10_A, Q10_B)
if Q10_X != "Matrix is singular":
np2latex(transpose(Q10_X), "X_{"+str(n)+"}")
print("")
def P8():
print("---------------------------------")
print("P8: Page 78 Problem 7")
Q8_A = np.array([[2, -1, 0, 0],
[0, 0, -1, 1],
[0, -1, 2, -1],
[-1, 2, -1, 0]], dtype=float)
Q8_B = np.array([1, 0, 0, 0], dtype=float)
np2latex(Q8_A, "A")
np2latex(transpose(Q8_B), "B")
Q8_X = gausPivot(Q8_A, Q8_B)
np2latex(transpose(Q8_X), "X")
def P9():
print("---------------------------------")
print("P9: Page 79 Problem 12")
#Use your program to solve the equations for k=10 and W=100.
k = 10
W = 100
k1, k2, k3, k4, k5 = k, 2*k, k, k, 2*k
W1, W2, W3 = 2*W, W, 2*W
Q9_A = np.array([[k1+k2+k3+k5, -k3, -k5],
[-k3, k3+k4, -k4],
[-k5, -k4, k4+k5]], dtype=float)
Q9_B = np.array([W1, W2, W3], dtype=float)
np2latex(Q9_A, "A")
np2latex(transpose(Q9_B), "B")
Q9_X = gausPivot(Q9_A, Q9_B)
np2latex(transpose(Q9_X), "X")
[-1.84, 3.03, 1.29],\
[-1.57, 5.25, 4.30]], dtype=float)
Q1c_A = np.array([[ 2, -1, 0],\
[-1, 2, -1],\
[ 0, -1, 2]], dtype=float)
Q1d_A = np.array([[4, 3, -1],\
[7, -2, 3],\
[5, -18, 13]], dtype=float)
# Using my functions
print("---------------------------------")
for A, part in zip([Q1a_A,Q1b_A,Q1c_A,Q1d_A],\
["a","b","c","d"]):
print("Question 1 P 55", part)
#print("A = \n",A)
np2latex(A, "A")
(LU, P) = LUdecomp(A)
#print("L + U - I = \n",LU)
np2latex(P, 'P') #print("P = \n",P)
(L, U) = LUexpand(LU)
np2latex(L, 'L') #print("L = \n",L)
np2latex(U, 'U') #print("U = \n",U)
#print("P^-1 * L * U =\n", np.dot(transpose(P),\
# np.dot(L,U)))
#Show that LU works!
np2latex(np.dot(transpose(P),np.dot(L,U)),\
'P^T\cdot L\cdot U')
(d, nonsing, cond) = LUnonsingular(A, LU, P)
print("Det: ", d)
print("Non-singular: ", nonsing)