2.8.3 Practice set: numerical calculus

Finite differences and their errors

Finite differences and their errors (lesson 2.8.1)

0 of 8 correct

1.
The forward difference is off by about $\frac{1}{2}f''(x)\,h$ . Estimate its error for $x^3$ at 2 with $h = 0.01$
2.
The backward difference is off by about $-\frac{1}{2}f''(x)\,h$ . Estimate its error for $\ln x$ at 1 with $h = 0.1$ , sign included
3.
The central difference is off by about $\frac{1}{6}f'''(x)\,h^2$ . Estimate its error for $e^x$ at 0 with $h = 0.1$
4.
Find the second difference $\frac{f(1.5) - 2f(1) + f(0.5)}{0.25}$ for $f(x) = x^4$ . The true $f''(1)$ is 12
5.
The best forward step is $\sqrt{2\varepsilon|f(x)|/|f''(x)|}$ . With $\varepsilon = 2 \times 10^{-16}$ , $|f(x)| = 2$ and $|f''(x)| = 8$ , it is $10^k$ . What is $k$ ?
6.
The default central step is $\varepsilon^{1/3}s \approx 6 \times 10^{-6}\,s$ . What step suits $x^5$ at $x = 200$ ?
7.
Find the central difference of $\max(x, 0)$ at $x = 0.01$ with $h = 0.05$ . The true slope there is 1
8.
Someone estimates $f''$ with $h = 10^{-8}$ , where $|f(x)| = 1$ . With $\varepsilon = 2.2 \times 10^{-16}$ , how big can the rounding error $\frac{2\varepsilon|f(x)|}{h^2}$ be?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Trapezoid and Simpson rules

Trapezoid and Simpson rules (lesson 2.8.2)

0 of 8 correct

1.
Use the trapezoid rule with 3 strips for $x^2$ on $[0, 3]$
2.
For $e^x$ on $[0, 1]$ with 4 strips, the left sum is 1.512437 and the right sum 1.942007. What is the trapezoid rule?
3.
Use $\frac{1}{12}\Delta x^2\,\big(f'(b) - f'(a)\big)$ to estimate the trapezoid error for $e^x$ on $[0, 2]$ with $\Delta x = 0.1$ (type e for $e$ )
4.
For a curve that bends down, a trapezoid sum is off by $-0.02$ . About how far off is the midpoint sum with the same strips? Include the sign
5.
With 2 strips, some integral has $M_2 = 1.2$ and $T_2 = 1.5$ . What is Simpson's rule with 4 strips?
6.
Use Simpson's rule with 2 strips for $4x^3 + 1$ on $[0, 1]$
7.
Use $\frac{1}{180}\Delta x^4\,\big(f'''(b) - f'''(a)\big)$ to find Simpson's error for $x^4$ on $[0, 2]$ with 4 strips
8.
Someone counts the end heights in full instead of halving them: for $x^2$ on $[0, 2]$ with 2 strips they get $\Delta x\,(0 + 1 + 4)$ . What do they get? (The trapezoid rule gives 3)

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

A derivative that picks its step, and Simpson's rule

Write three functions, using lessons 2.8.1 and 2.8.2. EPS in the starter is the machine epsilon $\varepsilon$ .

central_diff(f, x, h=None) returns the central difference $\frac{f(x + h) - f(x - h)}{2h}$ . When h is None, use the default step $\varepsilon^{1/3}\max(|x|, 1)$ .
second_diff(f, x, h=None) returns the second difference $\frac{f(x + h) - 2f(x) + f(x - h)}{h^2}$ . When h is None, use $\varepsilon^{1/4}\max(|x|, 1)$ .
simpson(f, a, b, n) returns Simpson's rule with n strips. Raise ValueError unless n is even and at least 2.

The tests include $\sqrt{x}$ at $10^6$ and $\ln x$ at $10^4$ , where a fixed step such as $10^{-5}$ fails. They also check that Simpson's error shrinks about 16 times when the strips double, and that a reversed interval gives a negative answer.

See the tests

import math


def test_central_exp():
    """With the default step, the slope of eˣ at 1 is e to about 10 digits"""
    expect_close(central_diff(math.exp, 1.0), math.e, what="central_diff(math.exp, 1.0)", rel=1e-9)


def test_central_given_step():
    """A step you pass is used as it is: x² at 3 with h = 0.5 gives exactly 6"""
    expect_close(central_diff(lambda x: x**2, 3.0, h=0.5), 6.0, what="central_diff(x², 3, h=0.5)", rel=1e-12)


def test_central_scales_with_x():
    """The default step grows with x: the slope of √x at 10⁶ is 0.0005 to about 10 digits"""
    expect_close(central_diff(math.sqrt, 1e6), 0.0005, what="central_diff(math.sqrt, 1e6)", rel=1e-10)


def test_central_at_zero():
    """At x = 0 the default step is still sensible: the slope of sin x at 0 is 1"""
    expect_close(central_diff(math.sin, 0.0), 1.0, what="central_diff(math.sin, 0.0)", rel=1e-9)


def test_second_exp():
    """With the default step, the second derivative of eˣ at 1 is e to about 7 digits"""
    expect_close(second_diff(math.exp, 1.0), math.e, what="second_diff(math.exp, 1.0)", rel=1e-7)


def test_second_scales_with_x():
    """The default second-difference step grows with x: ln x at 10⁴ has f'' = −10⁻⁸"""
    expect_close(second_diff(math.log, 1e4), -1e-8, what="second_diff(math.log, 1e4)", rel=1e-5)


def test_simpson_cubic_exact():
    """Simpson's rule is exact for a cubic: x³ on [0, 2] with 2 strips gives 4"""
    expect_close(simpson(lambda x: x**3, 0.0, 2.0, 2), 4.0, what="simpson(x³, 0, 2, 2)", rel=1e-12)


def test_simpson_recip():
    """1/x on [1, 2] with 10 strips gives 0.6931502307, close to ln 2"""
    expect_close(simpson(lambda x: 1 / x, 1.0, 2.0, 10), 0.6931502306889303, what="simpson(1/x, 1, 2, 10)", rel=1e-12)


def test_simpson_order():
    """Doubling the strips cuts the error about 16 times: eˣ on [0, 1] with 8 and 16 strips"""
    exact = math.e - 1
    e8 = abs(simpson(math.exp, 0.0, 1.0, 8) - exact)
    e16 = abs(simpson(math.exp, 0.0, 1.0, 16) - exact)
    assert 14 < e8 / e16 < 18, f"expected the error to shrink about 16 times, but it shrank {e8 / e16:.2f} times"


def test_simpson_reversed():
    """A reversed interval gives the negative: sin x from π to 0 with 20 strips is about −2"""
    expect_close(simpson(math.sin, math.pi, 0.0, 20), -2.0, what="simpson(sin, π, 0, 20)", rel=1e-5)


def test_simpson_odd():
    """An odd number of strips raises ValueError"""
    try:
        simpson(lambda x: x**2, 0.0, 1.0, 3)
    except ValueError:
        return
    raise AssertionError("expected ValueError for 3 strips: Simpson's rule pairs the strips")


def test_simpson_zero_strips():
    """Zero strips raises ValueError"""
    try:
        simpson(lambda x: x**2, 0.0, 1.0, 0)
    except ValueError:
        return
    raise AssertionError("expected ValueError for 0 strips")

Finite differences and their errors

Trapezoid and Simpson rules

Coding exercise

Free further reading