Birkhoff interpolation
E637944
Birkhoff interpolation is a generalized form of polynomial interpolation that allows prescribing function and derivative values at selected points, not necessarily in a consecutive or complete pattern.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
interpolation method
ⓘ
polynomial interpolation generalization ⓘ |
| allows |
mixed conditions of function and derivatives
ⓘ
prescribing derivative values at selected points ⓘ prescribing function values at selected points ⓘ |
| alsoKnownAs | lacunary interpolation ⓘ |
| appliesTo |
complex-valued functions
ⓘ
real-valued functions ⓘ |
| canBeFormulatedAs | linear system of equations for polynomial coefficients ⓘ |
| characteristic |
interpolation conditions need not be complete at each node
ⓘ
interpolation conditions need not be consecutive in derivative order ⓘ interpolation nodes may repeat with different derivative orders ⓘ |
| concerns |
distribution of derivative conditions among nodes
ⓘ
optimal placement of interpolation conditions ⓘ |
| contrastsWith | classical interpolation with complete derivative data at each node ⓘ |
| field |
approximation theory
ⓘ
computational mathematics ⓘ numerical analysis ⓘ |
| generalizes |
Hermite interpolation
NERFINISHED
ⓘ
Lagrange interpolation NERFINISHED ⓘ |
| goal | construct a polynomial matching given data with minimal degree ⓘ |
| hasVariant |
Hermite–Birkhoff surface interpolation
ⓘ
multivariate Birkhoff interpolation NERFINISHED ⓘ |
| involves |
derivative orders
ⓘ
interpolation nodes ⓘ linear conditions on polynomials ⓘ |
| mathematicalDomain |
algebra
ⓘ
analysis ⓘ |
| mayHave |
no solution for some interpolation schemes
ⓘ
non-unique solutions for some interpolation schemes ⓘ |
| namedAfter | George David Birkhoff NERFINISHED ⓘ |
| oftenAssumes |
finite set of derivative conditions
ⓘ
finite set of interpolation nodes ⓘ |
| relatedTo |
Hermite–Birkhoff interpolation
NERFINISHED
ⓘ
Vandermonde-type matrices ⓘ interpolation matrices ⓘ unisolvence in interpolation ⓘ |
| requires | differentiability of the target function up to prescribed orders ⓘ |
| studies |
degree bounds for interpolating polynomials
ⓘ
existence of interpolating polynomials ⓘ uniqueness of interpolating polynomials ⓘ |
| typicalProblem | find a polynomial satisfying prescribed function and derivative values at given points ⓘ |
| usedIn |
approximation of smooth functions
ⓘ
construction of special basis functions ⓘ numerical solution of differential equations ⓘ |
| uses | polynomials ⓘ |
Referenced by (1)
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