eval-certified-incl-error
▸ function evalCertifiedInclError
(p: number[][], x: number, pE?: number[], multiplier?: number): object
Defined in evaluate/double/eval-certified-incl-error.ts:57
Returns the result of evaluating the given polynomial (with specified
coefficient-wise error bounds) at x such that the sign is correct when
positive or negative and undecided when 0 - an additional multiplier
parameter can enforce additional bits (beyond the sign bit) to be correct.
- designed to be fast in 'easy' cases (say condition number < 2^53) and harder cases (condition number < 2^106) since nearly all typical calculations will have condition number < 2^106
- a staggered approach is used - first double precision, then simulated double-double precision (i.e. once compensated Horner evluation) is tried before giving up and returning 0 - see point below
- if zero is returned then the calculated result is too close to 0 to determine the sign; the caller of this function can then resort to a more accurate (possibly exact) evaluation
Parameters:
| Name | Type | Default value | Description |
|---|---|---|---|
p | number[][] | - | an array of 2 polynomials with coefficients given densely as an array of double precision floating point numbers from highest to lowest power, e.g. [5,-3,0] represents the polynomial 5x^2 - 3x; the first polynomial's coefficients represent the 'high part' (a double) of a double-double precision value, while the second polynomial's coefficients represent the 'low part', i.e. designating hp for high part and lp for low part it must be that they are non-overlapping -> twoSum(lp,hp) will equal [lp,hp]; put another way, if the given polynomial is given as e.g. a linear polynomial with coefficients in double precision, e.g. [[1.7053025658242404e-13, 2354.33721613], [-7.105427357601002e-15,284.5673337]] then this parameter, p, should be [[2354.33721613], 284.5673337], [1.7053025658242404e-13, -7.105427357601002e-15]] which is simply the result of transposing the original polynomial if it is seen as a matrix |
x | number | - | the value at which to evaluate the polynomial |
pE | number[] | undefined | defaults to undefined; an error polynomial that provides a coefficient-wise error bound on the input polynomial; all coefficients must be positive; if undefined then the input polynomial will be assumed exact |
multiplier | number | 1 | defaults to 1; the final calculation error needs to be a multiple of this number smaller than the evaluated value, otherwise zero is returned - useful if not only the sign is important but also some bits, e.g. if multiplier = 8 then 3 bits will have to be correct otherwise 0 is returned |
Returns: object
| Name | Type |
|---|---|
r̂ | number |
e | number |