def opt_prob_1c():
f1 = lambda x, y: 3*x - 2*y
#find corner points:
ln1 = make_line_eq(make_plus(make_var('x'),make_var('y')), make_const(0.0))# x+y=0
ln2 = make_line_eq(make_var('x'),make_var('y'))# x-y=0 => x=y
ln3 = make_line_eq(make_plus(make_prod(make_const(-2.0), make_var('x')),make_prod(make_const(4.0), make_var('y'))), make_const(5.0))# -2x+4y=5
p12 = line_intersection(ln1, ln2)
p13 = line_intersection(ln1, ln3)
p23 = line_intersection(ln2, ln3)
possible_cps = [p12,p13,p23]
corner_points = [pt for pt in possible_cps if -2*pt.get_x().get_val() + 4*pt.get_y().get_val() <= 5] # -2x+4y<=5
corner_points = [pt for pt in corner_points if pt.get_x().get_val() + pt.get_y().get_val() >= 0] #x+y>=0
corner_points = [pt for pt in corner_points if pt.get_x().get_val() - pt.get_y().get_val() <= 0] #x-y<=0
# for pt in corner_points:
# print(pt.get_x().get_val(), pt.get_y().get_val())
print '1c: ',maximize_obj_fun(f1, corner_points)
def test_11(self):
'''
(x+1)^4 *(4x-1)^2 => (x+1)^4 *(4x-1)^2 * ( (4/(x+1) + 8/(4x-1))
ln(x+1)^4
'''
print('*******Test 10********')
fex1 = make_pwr_expr(make_plus(make_pwr('x', 1.0), make_const(1.0)), 4.0)
fex2 = make_pwr_expr(make_plus(make_prod(make_const(4.0),
make_pwr('x', 1.0)),
make_const(-1.0)), 2.0)
fex = make_prod(fex1, fex2)
print(fex)
drv = logdiff(fex)
assert not drv is None
print(drv)
drvf = tof(drv)
assert not drvf is None
def gt_drvf(x):
z1 = ((x + 1.0) **4.0) * ((4*x - 1.0)** 2.0)
z2 = (4.0/(x + 1.0)) + (8.0/ (4*x - 1.0))
return z1 * z2
err = 0.0001
for i in range(1, 10):
#print(drvf(i), gt_drvf(i))
assert abs(gt_drvf(i) - drvf(i)) <= err
print('Test 11: pass')
def test_assign_03_prob_01_ut_04(self):
print('\n***** Assign 03: Problem 01: Unit Test 04 *****')
e1 = make_pwr('x', 2.0)
e2 = make_plus(e1, make_prod(make_const(0.0), make_pwr('x', 1.0)))
e3 = make_plus(e2, make_const(-1.0))
e4 = make_pwr_expr(e3, 4.0)
e5 = make_pwr('x', 2.0)
e6 = make_plus(e5, make_prod(make_const(0.0), make_pwr('x', 1.0)))
e7 = make_plus(e6, make_const(1.0))
e8 = make_pwr_expr(e7, 5.0)
e9 = make_prod(e4, e8)
print(e9)
e9f = tof(deriv(e9))
assert not e9f is None
err = 0.000001
def drvf(x):
return 2 * x * ((x**2 - 1.0)**3) * (
(x**2 + 1.0)**4) * (9 * x**2 - 1.0)
for i in range(10):
#print(e9f(i), drvf(i))
assert abs(e9f(i) - drvf(i)) <= err
print('Assign 03: Problem 01: Unit Test 04: pass')
def test_prob_02_ut_01(self):
print('\n***** Problem 02: UT 01 ************')
fex = make_prod(make_pwr('x', 1.0),
make_prod(make_plus(make_pwr('x', 1.0),
make_const(1.0)),
make_plus(make_pwr('x', 1.0),
make_const(2.0))))
drv = logdiff(fex)
assert not drv is None
print(drv)
drvf = tof(drv)
assert not drvf is None
def gt_drvf(x):
z = x*(x + 1.0)*(x + 2.0)
z2 = (1.0/x + 1.0/(x + 1.0) + 1.0/(x + 2.0))
return z * z2
err = 0.0001
for i in range(1, 10):
#print(drvf(i), gt_drvf(i))
assert abs(gt_drvf(i) - drvf(i)) <= err
for i in range(-10, -1):
if i == -1 or i == -2:
continue
#print(drvf(i), gt_drvf(i))
assert abs(gt_drvf(i) - drvf(i)) <= err
print('Problem 02: UT 01: pass')
def test_assign_03_prob_01_ut_05(self):
print('\n***** Assign 03: Problem 01: Unit Test 05 *****')
e1 = make_plus(make_pwr('x', 1.0), make_const(1.0))
e2 = make_pwr('x', 3.0)
e3 = make_prod(make_const(0.0), make_pwr('x', 2.0))
e4 = make_plus(e2, e3)
e5 = make_prod(make_const(5.0), make_pwr('x', 1.0))
e6 = make_plus(e4, e5)
e7 = make_plus(e6, make_const(2.0))
e8 = make_prod(e1, e7)
# 1) print the expression we just constructed
print('-- function expression is:\n')
print(e8)
# 2) differentiate and make sure that it not None
drv = deriv(e8)
assert not drv is None
print('\n-- derivative is:\n')
print(e8)
# 3) convert drv into a function
e8f = tof(drv)
assert not e8f is None
# steps 2) and 3) can be combined into tof(deriv(e6)).
# 4) construct the ground truth function
gt = lambda x: 4.0 * (x**3) + 3 * (x**2) + 10.0 * x + 7.0
# 5) compare the ground gruth with what we got in
# step 3) on an appropriate number range.
print('\n--comparison with ground truth:\n')
err = 0.00001
for i in range(15):
#print(e8f(i), gt(i))
assert abs(e8f(i) - gt(i)) <= err
print('Assign 03: Problem 01: Unit Test 05: pass')
def solve_pdeq(k1, k2):
#k1*y' = k2*y
assert isinstance(k1, const)
assert isinstance(k2, const)
return make_prod(
make_const(1.0),
make_e_expr(make_prod(make_quot(k2, k1), make_pwr('t', 1.0))))
def test_06(self):
#x^-3+7e^(5x)+4*x^-1 => (1/-2)*(x^-2)+0 +(7*(1/5)*e(5x)+ (4*ln|x|)
print("****Unit Test 06********")
fex1 = make_pwr('x', -3.0)
fex2 = make_prod(
make_const(7.0),
make_e_expr(make_prod(make_const(5.0), make_pwr('x', 1.0))))
fex3 = make_prod(make_const(4.0), make_pwr('x', -1.0))
fex4 = make_plus(fex1, fex2)
fex = make_plus(fex4, fex3)
print(fex)
afex = antideriv(fex)
assert not afex is None
def gt(x):
v1 = -0.5 * (x**(-2.0))
v2 = (7.0 / 5.0) * (math.e**(5.0 * x))
v3 = 4.0 * (math.log(abs(x), math.e))
return v1 + v2 + v3
afexf = tof(afex)
assert not afexf is None
err = 0.001
for i in range(1, 10):
print(afexf(i), gt(i))
assert abs(afexf(i) - gt(i)) <= err * gt(i)
print("Unit Test 06 pass")
def test_07():
ln1 = make_line_eq(make_var('y'), make_plus(make_prod(make_const(-1.0/5.0),
make_pwr('x', 1.0)),
make_const(10.0)))
ln2 = make_line_eq(make_var('y'), make_plus(make_prod(make_const(1.0/5.0),
make_pwr('x', 1.0)),
make_const(5.0)))
print(line_intersection(ln1, ln2))
def nra_ut_13():
#approximate a zero for x^3-3x^2+14x-2
f1 = make_plus(make_pwr('x', 3.0),
make_prod(const(-3.0), make_pwr('x', 2.0)))
f2 = make_plus(make_prod(const(14.0), make_pwr('x', 1.0)), const(-2.0))
fexpr = make_plus(f1, f2)
approx = nra(fexpr, const(1.0), const(1000000))
print(approx)
def test_11():
ln1 = make_line_eq(make_var('x'), make_const(1.0))
ln2 = make_line_eq(make_var('y'), make_prod(make_const(0.5),
make_pwr('x', 1.0)))
print(line_intersection(ln1, ln2))
ln3 = make_line_eq(make_var('y'), make_plus(make_prod(make_const(-3.0/4),
make_pwr('x', 1.0)),
make_const(3.0)))
print(line_intersection(ln1, ln3))
print(line_intersection(ln2, ln3))
def percent_retention_model(lmbda, a):
#r(t) = (100-a)e^(-lmbda*t) + a
assert isinstance(lmbda, const)
assert isinstance(a, const)
left = const(100.0 - a.get_val())
right = make_e_expr(
make_prod(const(-1.0 * lmbda.get_val()), make_pwr('t', 1.0)))
return make_plus(make_prod(left, right), a)
def spread_of_news_model(p, k):
assert isinstance(p, const) and isinstance(k, const)
expon = const(-1.0 * k.get_val())
return make_prod(
p,
make_plus(
const(1.0),
make_prod(const(-1.0),
make_e_expr(make_prod(expon, make_pwr('t', 1.0))))))
def max_rev_test():
print("***Max Revenue Test 01 *****")
e1 = make_prod( make_const(1.0/12.0), make_pwr('x', 2.0))
e2 = make_prod(make_const(-10.0), make_pwr('x', 1.0))
sum1 = make_plus(e1, e2)
demand_expr = make_plus(sum1, make_const(300.0))
num_units, rev, price = maximize_revenue(demand_expr, constraint=lambda x: 0 <= x <= 60)
print('x = ', num_units.get_val())
print("rev= ", rev.get_val())
print('price = ', price.get_val())
print("Max Revenue Test: pass")
def test_03():
f1 = make_prod(make_const(1.0 / 3.0), make_pwr('x', 3.0))
f2 = make_prod(make_const(-2.0), make_pwr('x', 2.0))
f3 = make_prod(make_const(3.0), make_pwr('x', 1.0))
f4 = make_plus(f1, f2)
f5 = make_plus(f4, f3)
poly = make_plus(f5, make_const(1.0))
print 'f(x) = ', poly
xtrma = loc_xtrm_1st_drv_test(poly)
for i, j in xtrma:
print i, str(j)
def test_05(): #y = x; y = 2x; y = 3x-10
ln1 = make_line_eq(make_var('y'), make_pwr('x', 1.0))
ln2 = make_line_eq(make_var('y'),
make_prod(make_const(2.0), make_pwr('x', 1.0)))
ln3 = make_line_eq(
make_var('y'),
make_plus(make_prod(make_const(3.0), make_pwr('x', 1.0)),
make_const(-10.0)))
print(get_line_coeffs(ln1))
print(get_line_coeffs(ln2))
print(get_line_coeffs(ln3))
def test_02():
f0 = make_prod(make_const(0.5), make_pwr('x', 2.0))
f1 = make_prod(make_const(6.0), make_pwr('x', 1.0))
f2 = make_plus(f0, f1)
poly = make_plus(f2, make_const(0.0))
print '+++', poly
zeros = find_poly_2_zeros(poly)
for c in zeros:
print c
pf = tof(poly)
for c in zeros:
assert abs(pf(c.get_val()) - 0.0) <= 0.0001
print 'True zeros!'
def test_assign_03_prob_01_ut_06(self):
print('\n***** Assign 03: Problem 01: Unit Test 06 *****')
e1 = make_prod(make_const(2.0), make_pwr('x', 4.0))
e2 = make_prod(make_const(-1.0), make_pwr('x', 1.0))
e3 = make_plus(e1, e2)
e4 = make_plus(e3, make_const(1.0))
e5 = make_prod(make_const(-1.0), make_pwr('x', 5.0))
e6 = make_prod(make_const(0.0), make_pwr('x', 4.0))
e7 = make_plus(e5, e6)
e8 = make_prod(make_const(0.0), make_pwr('x', 3.0))
e9 = make_plus(e7, e8)
e10 = make_prod(make_const(0.0), make_pwr('x', 2.0))
e11 = make_plus(e9, e10)
e12 = make_prod(make_const(0.0), make_pwr('x', 1.0))
e13 = make_plus(e11, e12)
e14 = make_plus(e13, make_const(1.0))
e15 = make_prod(e4, e14)
print('-- function expression is:\n')
print(e15)
drv = deriv(e15)
assert not drv is None
print('\n-- derivative is:\n')
print(drv)
e15f = tof(drv)
assert not e15f is None
gt = lambda x: -18.0 * (x**8) + 6.0 * (x**5) - 5.0 * (x**4) + 8.0 * (
x**3) - 1.0
err = 0.00001
print('\n--comparison with ground truth:\n')
for i in range(10):
#print(e15f(i), gt(i))
assert abs(e15f(i) - gt(i)) <= err
print('Assign 03: Problem 01: Unit Test 06: pass')
def problem_1_deriv(): # still has problems
# f(x) =(x+1)(2x+1)(3x+1) /(4x+1)^.5
fex1 = make_plus(make_pwr('x', 1.0), const(1.0))
fex2 = make_plus(make_prod(const(2.0), make_pwr('x', 1.0)), const(1.0))
fex3 = make_plus(make_prod(const(3.0), make_pwr('x', 1.0)), const(1.0))
fex4 = make_prod(fex1, fex2)
fex5 = make_prod(fex4, fex3)
fex6 = make_pwr_expr(
make_plus(make_prod(const(4.0), make_pwr('x', 1.0)), const(1.0)), 0.5)
fex = make_quot(fex5, fex6)
print(fex)
drv = deriv(fex)
print('drv: ', drv)
drvf = tof(drv)
def gt_drvf(x):
n = (60.0 * x**3) + (84 * x**2) + 34 * x + 4
d = (4 * x + 1)**(3 / 2)
return n / d
for i in range(1, 10):
print(drvf(i), gt_drvf(i))
assert abs(gt_drvf(i) - drvf(i)) <= 0.001
assert drv is not None
print(drv)
# zeros = find_poly_2_zeros(drv)
# print(zeros)
f1 = tof(fex)
f2 = tof(deriv(fex))
xvals = np.linspace(1, 5, 10000)
yvals1 = np.array([f1(x) for x in xvals])
yvals2 = np.array([f2(x) for x in xvals])
fig1 = plt.figure(1)
fig1.suptitle('Graph')
plt.xlabel('x')
plt.ylabel('y')
plt.ylim()
plt.xlim()
plt.grid()
plt.plot(xvals, yvals1, label=fex.__str__(), c='r')
plt.plot(xvals, yvals2, label=deriv(fex).__str__(), c='g')
plt.legend(loc='best')
plt.show()
def test_05():
ln1 = make_line_eq(make_var('y'), make_pwr('x', 1.0))
ln2 = make_line_eq(make_var('y'), make_prod(make_const(2.0),
make_pwr('x', 1.0)))
ln3 = make_line_eq(make_var('y'), make_plus(make_prod(make_const(3.0),
make_pwr('x', 1.0)),
make_const(-10.0)))
print(get_line_coeffs(ln1))
print(get_line_coeffs(ln2))
print(get_line_coeffs(ln3))
print(line_intersection(ln1, ln2))
print(line_intersection(ln2, ln3))
print(line_intersection(ln1, ln3))
def oil_disk_test():
yt = make_prod(make_const(0.02 * math.pi), make_pwr('r', 2.0))
print(yt)
dydt = dydt_given_x_dxdt(yt, make_const(150.0), make_const(20.0))
assert not dydt is None
assert isinstance(dydt, const)
print(dydt)
def test_10(self):
#(3x+2)^4
print("****Unit Test 10********")
fex0 = make_prod(make_const(3.0), make_pwr('x', 1.0))
fex1 = make_plus(fex0, make_const(2.0))
fex = make_pwr_expr(fex1, 4.0)
print(fex)
afex = antideriv(fex)
assert not afex is None
print(afex)
afexf = tof(afex)
err = 0.0001
def gt(x):
return (1.0 / 15) * ((3 * x + 2.0)**5)
for i in range(1, 10):
assert abs(afexf(i) - gt(i)) <= err
fexf = tof(fex)
assert not fexf is None
fex2 = deriv(afex)
assert not fex2 is None
print(fex2)
fex2f = tof(fex2)
assert not fex2f is None
for i in range(1, 1000):
assert abs(fexf(i) - fex2f(i)) <= err
print('Test 10:pass')
def test_04():
ln1 = make_line_eq(make_var('y'), make_const(2.0))
ln2 = make_line_eq(make_var('y'), make_plus(make_prod(make_const(2.0),
make_pwr('x', 1.0)),
make_const(-6.0)))
print(line_intersection(ln1, ln2))
print(line_intersection(ln2, ln1))
def test_05():
fexpr2_01 = make_prod(make_pwr('x', 1.0), make_e_expr(make_pwr('x', 1.0)))
print(gt21_02(2.001))
fex = taylor_poly(fexpr2_01, make_const(2.001), make_const(2))
print(fex)
fex_tof = tof(fex)
print(fex_tof(2.001))
def gt21_02(x):
''' ground truth for 2nd taylor of fexpr2_01. '''
fexpr2_01 = make_prod(make_pwr('x', 1.0), make_e_expr(make_pwr('x', 1.0)))
f0 = tof(fexpr2_01)
f1 = tof(deriv(fexpr2_01))
f2 = tof(deriv(deriv(fexpr2_01)))
return f0(2.0) + f1(2.0) * (x - 2.0) + (f2(2.0) / 2) * (x - 2.0)**2
def test_11():
f1 = make_prod(make_const(-1.0), make_pwr('x', 3.0))
f2 = make_prod(make_const(8.5), make_pwr('x', 2.0))
f3 = make_prod(make_const(0.0), make_pwr('x', 0.0))
f4 = make_plus(f1, f2)
f5 = make_plus(f4, f3)
f6 = make_plus(f5, make_const(100.0))
print(f6)
ips = find_infl_pnts(f6)
err = 0.0001
assert len(ips) == 1
ip = ips[0]
# assert abs(ip.get_x().get_val() - 1.0) <= err
# assert abs(ip.get_y().get_val() - 3.0) <= err
print("inflection points: ", ip)
def test_12():
ln1 = make_line_eq(make_var('x'), make_const(0.0))
ln2 = make_line_eq(make_var('y'), make_const(0.0))
ln3 = make_line_eq(make_var('y'), make_plus(make_prod(make_const(-4.0/3),
make_pwr('x', 1.0)),
make_const(160.0)))
ln4 = make_line_eq(make_var('y'), make_plus(make_prod(make_const(-0.5),
make_pwr('x', 1.0)),
make_const(120.0)))
print(ln1)
print(ln3)
print(line_intersection(ln1, ln3))
print(ln2)
print(ln3)
print(line_intersection(ln2, ln3))
print(line_intersection(ln3, ln4))
def nra_ut_00():
''' Approximating x^3 - x - 2. '''
fexpr = make_pwr('x', 3.0)
fexpr = make_plus(fexpr, make_prod(make_const(-1.0), make_pwr('x', 1.0)))
fexpr = make_plus(fexpr, make_const(-2.0))
# print('fexpr',str(fexpr))
print(nra(fexpr, make_const(1.0), make_const(10)))
def test_09(): #y = 5; y = -x +6
ln1 = make_line_eq(make_var('y'), make_const(5.0))
ln2 = make_line_eq(
make_var('y'),
make_plus(make_prod(make_const(-1.0), make_pwr('x', 1.0)),
make_const(6.0)))
print(line_intersection(ln1, ln2))
def test_09(self):
#3*(x+2)^-1 => 3*ln|x+2|
print("****Unit Test 09********")
fex0 = make_plus(make_pwr('x', 1.0), make_const(2.0))
fex1 = make_pwr_expr(fex0, -1.0)
fex = make_prod(make_const(3.0), fex1)
print(fex)
afex = antideriv(fex)
print("antideriv: ", afex)
err = 0.0001
afexf = tof(afex)
def gt(x):
return 3.0 * math.log(abs(2.0 + x), math.e)
for i in range(1, 101):
assert abs(afexf(i) - gt(i)) <= err
assert not afex is None
print(afex)
fexf = tof(fex)
assert not fexf is None
fex2 = deriv(afex)
assert not fex2 is None
print(fex2)
fex2f = tof(fex2)
assert not fex2f is None
for i in range(1, 1000):
assert abs(fexf(i) - fex2f(i)) <= err
print('Test 09:pass')
def arm_tumor_test():
yt = make_prod(make_const(0.003 * math.pi), make_pwr('r', 3.0))
print(yt)
dydt = dydt_given_x_dxdt(yt, make_const(10.3), make_const(-1.75))
assert not dydt is None
assert isinstance(dydt, const)
print(dydt)