calc: Interpolation
11.8.6 Polynomial Interpolation
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The ‘a p’ (‘calc-poly-interp’) [‘polint’] command does a polynomial
interpolation at a particular ‘x’ value. It takes two arguments from
the stack: A data matrix of the sort used by ‘a F’, and a single number
which represents the desired ‘x’ value. Calc effectively does an exact
polynomial fit as if by ‘a F i’, then substitutes the ‘x’ value into the
result in order to get an approximate ‘y’ value based on the fit. (Calc
does not actually use ‘a F i’, however; it uses a direct method which is
both more efficient and more numerically stable.)
The result of ‘a p’ is actually a vector of two values: The ‘y’ value
approximation, and an error measure ‘dy’ that reflects Calc’s estimation
of the probable error of the approximation at that value of ‘x’. If the
input ‘x’ is equal to any of the ‘x’ values in the data matrix, the
output ‘y’ will be the corresponding ‘y’ value from the matrix, and the
output ‘dy’ will be exactly zero.
A prefix argument of 2 causes ‘a p’ to take separate x- and y-vectors
from the stack instead of one data matrix.
If ‘x’ is a vector of numbers, ‘a p’ will return a matrix of
interpolated results for each of those ‘x’ values. (The matrix will
have two columns, the ‘y’ values and the ‘dy’ values.) If ‘x’ is a
formula instead of a number, the ‘polint’ function remains in symbolic
form; use the ‘a "’ command to expand it out to a formula that describes
the fit in symbolic terms.
In all cases, the ‘a p’ command leaves the data vectors or matrix on
the stack. Only the ‘x’ value is replaced by the result.
The ‘H a p’ [‘ratint’] command does a rational function
interpolation. It is used exactly like ‘a p’, except that it uses as
its model the quotient of two polynomials. If there are ‘N’ data
points, the numerator and denominator polynomials will each have degree
‘N/2’ (if ‘N’ is odd, the denominator will have degree one higher than
the numerator).
Rational approximations have the advantage that they can accurately
describe functions that have poles (points at which the function’s value
goes to infinity, so that the denominator polynomial of the
approximation goes to zero). If ‘x’ corresponds to a pole of the fitted
rational function, then the result will be a division by zero. If
Infinite mode is enabled, the result will be ‘[uinf, uinf]’.
There is no way to get the actual coefficients of the rational
function used by ‘H a p’. (The algorithm never generates these
coefficients explicitly, and quotients of polynomials are beyond ‘a F’’s
capabilities to fit.)
Info Catalog
calc: Curve Fitting Details
calc: Curve Fitting