1.6.3 Practice set: interpolation

Linear interpolation

Linear interpolation (lesson 1.6.1)

0 of 8 correct

1.
$f(0) = 2$ and $f(4) = 10$ . Estimate $f(1)$ by linear interpolation
2.
$x = 7$ lies between $x_0 = 5$ and $x_1 = 10$ , where the values are $y_0 = 50$ and $y_1 = 100$ . Interpolate at 7
3.
$f(1) = 9$ and $f(3) = 1$ . Estimate $f(2.5)$
4.
The known points are $(0, 4)$ , $(1, 6)$ and $(4, 3)$ . Estimate the value at $x = 3$
5.
$f(2) = 5$ and $f(4) = 9$ . At $x = 6$ , what weight does the line put on $y_0 = 5$ ?
6.
The known points are $(0, 4)$ , $(1, 6)$ and $(4, 3)$ . What does np.interp return at $x = 7$ ?
7.
A quantity is 2 at $x = 0$ and 32 at $x = 4$ . Interpolate in logs to estimate it at $x = 1$
8.
A car averages 50 km/h over its first 2 hours and 65 km/h over its first 4. Interpolate the distances: what is its average over the first 3 hours, in km/h?

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.

Cubic splines

Cubic splines (lesson 1.6.2)

0 of 8 correct

1.
A cubic spline through 9 points has how many cubic pieces?
2.
What is the highest degree of the one polynomial through 12 points with different $x$ values?
3.
The first piece of the natural spline through $(0, 0)$ , $(1, 1)$ and $(2, 0)$ is $1.5x - 0.5x^3$ . Evaluate it at $x = 0.2$
4.
Find the slope of the chord of $1.5x - 0.5x^3$ from 0.99 to 1
5.
The chain of lines through $(0, 0)$ , $(2, 4)$ and $(3, 3)$ has a corner at $x = 2$ . By how much does the slope change there?
6.
A spline through 7 points has 24 coefficients. Passing through the points and smooth joins fix some of them. How many are left for the end conditions?
7.
With 3 points, SciPy's default spline is one parabola. The points $(0, 0)$ , $(1, 1)$ and $(2, 4)$ lie on $x^2$ . What does the default spline give at $x = 1.5$ ?
8.
Data values lie between 3 and 7. What is the largest value linear interpolation can give between the points?

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

Linear interpolation from scratch

Write interpolate(xs, ys, x) that returns the linear interpolation of the points (xs[i], ys[i]) at a single number x. The xs are in increasing order, but not always evenly spaced.

Find the two neighbours with xs[i] <= x <= xs[i + 1], then use the formula from lesson 1.6.1 with that gap's own width.
If x is outside [xs[0], xs[-1]], raise ValueError: refuse to extrapolate, unlike np.interp.

The tests compare your answers with np.interp inside the data.

See the tests

import numpy as np

XS = [0, 1, 2, 4]
YS = [1, 3, 2, 6]


def test_between_two_points():
    """Halfway between (0, 1) and (1, 3) gives 2"""
    expect_close(interpolate(XS, YS, 0.5), 2.0, what="interpolate at 0.5")


def test_picks_the_right_gap():
    """At x = 3 it uses (2, 2) and (4, 6), giving 4"""
    expect_close(interpolate(XS, YS, 3), 4.0, what="interpolate at 3")


def test_at_known_points():
    """At a known point it returns that point's value"""
    for x, y in zip(XS, YS):
        expect_close(interpolate(XS, YS, x), y, what=f"interpolate at {x}")


def test_matches_numpy():
    """Agrees with np.interp at many points inside the data"""
    for x in np.linspace(0, 4, 37):
        expect_close(interpolate(XS, YS, x), np.interp(x, XS, YS), what=f"interpolate at {x:.3f}")


def test_refuses_to_extrapolate():
    """Outside the data, raise ValueError"""
    for x in (-1, 5):
        try:
            interpolate(XS, YS, x)
        except ValueError:
            continue
        raise AssertionError(f"expected ValueError at x = {x}")

Linear interpolation

Cubic splines

Coding exercise

Free further reading