Homework09
A MathWikiből
(Változatok közti eltérés)
(Új oldal, tartalma: „==Numerical Integral== write a function called num_int that calculates the numerical integral of sin(x)/x using trapezoidal rule. *The function should have three parame…”) |
(→Numerical Differential) |
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| (egy szerkesztő 2 közbeeső változata nincs mutatva) | |||
| 1. sor: | 1. sor: | ||
==Numerical Integral== | ==Numerical Integral== | ||
| − | + | Write a function called num_int that calculates the numerical integral of sin(x)/x using trapezoidal rule. | |
*The function should have three parameters: a, b and x | *The function should have three parameters: a, b and x | ||
*The endpoints of the interval to integrate over: 0 < a < b | *The endpoints of the interval to integrate over: 0 < a < b | ||
| 20. sor: | 20. sor: | ||
Hint: | Hint: | ||
There are one less trapezoids in the partition than points. " | There are one less trapezoids in the partition than points. " | ||
| + | |||
| + | ==Numerical Differential== | ||
| + | Write a function called num_diff that calculates the numerical derivative of a function using central difference. | ||
| + | *The function should have 2 parameters: y and x | ||
| + | *y will be a numpy array containing the values of the function (y-coordinates). | ||
| + | *x This will determine the x-coordinate values. | ||
| + | *Its default value should be None in which case the x values should be (1, 2, ..., n) (n is the length of y). Otherwise, this parameter will be a numpy array with the same length as y and a (xi, yi) pair represents a point on the graph of the function. | ||
| + | *Return a numpy array that contains the central differences and is two elements shorter than y. | ||
| + | Hint: | ||
| + | |||
| + | The formula for central difference: | ||
| + | |||
| + | f '(xi) ≈ (yi+1-yi-1) / (xi+1-xi-1) | ||
| + | If x=None then the denominator is constant 2. | ||
| + | This formula is invalid for the first and last i indexes, therefore the result is 2 elements shorted then the input: | ||
| + | (f '(x2), f '(x3), ... f '(xn-1)) | ||
| + | Use numpy! | ||
| + | |||
| + | Example | ||
| + | |||
| + | num_diff([1,4,9,16]) -> [4,6] | ||
| + | Which is the same as: | ||
| + | num_diff([1,4,9,16], [1,2,3,4]) -> [4,6] | ||
| + | In formula: [(9-1)/(3-1), (16-4)/(4-2)] | ||
| + | |||
| + | ==Note:== | ||
| + | Each problem counts 1 point | ||
A lap jelenlegi, 2021. május 5., 17:28-kori változata
Numerical Integral
Write a function called num_int that calculates the numerical integral of sin(x)/x using trapezoidal rule.
- The function should have three parameters: a, b and x
- The endpoints of the interval to integrate over: 0 < a < b
- The past parameter will be either a positive integer or a numpy array. This will represent the partition of the interval [a, b].
- Return the numerical approximation of the integral (a real number).
If the last parameter is a number, then divide the interval to that many trapezoids with equal width (use linspace).
If it is a numpy array, that will contain the points in the partition. The first number will be a and the last one b and the points will be increasing.
Example
Suppose that f(x) = sin(x)/x. num_int(1, 2, 1) Then the result should be: (f(1) + f(2))/2 num_int(1, 2, 2) Then the result should be: (f(1) + f(1.5))/4 + (f(1.5) + f(2))/4 Since the partition is: [1, 1.5, 2] In fact, the latter should be the same as: num_int(1, 2, numpy.array([1,1.5,2]))
Hint: There are one less trapezoids in the partition than points. "
Numerical Differential
Write a function called num_diff that calculates the numerical derivative of a function using central difference.
- The function should have 2 parameters: y and x
- y will be a numpy array containing the values of the function (y-coordinates).
- x This will determine the x-coordinate values.
- Its default value should be None in which case the x values should be (1, 2, ..., n) (n is the length of y). Otherwise, this parameter will be a numpy array with the same length as y and a (xi, yi) pair represents a point on the graph of the function.
- Return a numpy array that contains the central differences and is two elements shorter than y.
Hint:
The formula for central difference:
f '(xi) ≈ (yi+1-yi-1) / (xi+1-xi-1) If x=None then the denominator is constant 2. This formula is invalid for the first and last i indexes, therefore the result is 2 elements shorted then the input: (f '(x2), f '(x3), ... f '(xn-1)) Use numpy!
Example
num_diff([1,4,9,16]) -> [4,6] Which is the same as: num_diff([1,4,9,16], [1,2,3,4]) -> [4,6] In formula: [(9-1)/(3-1), (16-4)/(4-2)]
Note:
Each problem counts 1 point
