If the recording time is not neat, can I still fill in the data in between?
There is no need to forcibly move the record to the hour, so that every real input position can be included in the calculation.
General Algorithms and Numerical Computing
Sometimes you don't record at the same time, and the abscissa of the table may not be sorted. First put them into the graph as they are, and then use Lagrange interpolation; if it happens to be the midpoint of four equally spaced data, there is a simple special formula.
Guess first
Does Lagrange interpolation require records to be recorded at certain equal intervals?
Show answer and explanation
Not required, but the abscissa cannot be repeated — It directly uses each record's own abscissa, allowing unequal spacing and disordering.
Then connect experience and science
Each record retains its own abscissa
The spacing between points is different, which is not an error; there is no need to sort the input list first.
First read it as different xᵢ and the corresponding yᵢ
Construct a contribution for each point
Each basis function is equal to 1 at its own point and equal to 0 at other known points; after weighted addition, it passes through all known points.
y(x)=Σ yᵢ∏(x−xⱼ)/(xᵢ−xⱼ)
Equally spaced midpoints can be simplified
When four points are equally spaced, the two middle values have a weight of 9, and the outer values have a weight of −1, divided by 16. Not a common average.
y中 = [9(y₂+y₃)−y₁−y₄]/16
Repeated abscissas must be processed first
The same x cannot be subtracted from 0 in the denominator. If the observations give different readings at the same time, you need to decide how to deal with the errors first, and interpolation cannot be applied rigidly.
All xᵢ must be distinct from each other
For each i, the product goes over all j≠i, and the individual yᵢ weighted terms are added. All xᵢ must be different from each other and have the same unit; the four-point midpoint formula is only applicable to its specific equal interval conditions and cannot be used for any scattered points.
give it a try
30° of an unequal sine corresponds to about 43.097% of the position; 50% of the lunar surface is exactly 11h TD, which can be compared with the general interpolation and half-interval formulas.
No need to memorize symbols
- xᵢ / yᵢ
- Each known input and output
- Lᵢ(x)
- Lagrange basis polynomials
- y(x)
- Interpolation polynomial through all given points
Algorithm, accuracy and scope of application
Supplementary Chapter 3 3.12 and 3.13: The midpoint weight of four equally spaced samples is (-1,9,9,-1)/16; the unequal Lagrange item-by-item product method allows the input to be out of order, and the x values must be different from each other. The sine table comes from the six-point exercise on printed page 34. The query for 30° is about 0.5; the lunar right ascension is from Example 3.f. When the y unit is seconds relative to 10 hours, the query for 11h yields 1540.0014375s (rounded to 1540.001s in the book). The interface only queries within the given data range and does not extrapolate high-order polynomials as reliable predictions; cubic polynomials in 3.g of the original book are also used for independent verification. No userform input is provided, and the core pure function accepts unequal spaced data and rejects duplicate x's.
my understanding check
If the input list is not sorted by x, is the Lagrange result necessarily wrong?
Won't be wrong just because of disorder — The interpolation weight is calculated from the actual coordinates of each point, regardless of the input order.
Two points have the same x but different y. Can we directly apply the formula?
Can't — The denominator will be zero, and the observational implications of repeated abscissas should be resolved first.
Does passing all samples ensure that distant extrapolation is accurate?
No guarantee — High-order interpolating polynomials do not automatically provide the true physical laws at a distance.
The true position of each record is preserved; unequal intervals can be interpolated, but repeated x and long-distance extrapolation require additional processing.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 3 chapter
General Algorithms and Numerical Computing · All interactive courses