Pv loop experiments: Difference between revisions
Jump to navigation
Jump to search
en>Rjwilmsi m →References: Journal cites, added 1 DOI, using AWB (8073) |
en>BattyBot Converted {{Multiple issues}} to new format to fix expert parameter & general fixes using AWB (8853) |
||
| Line 1: | Line 1: | ||
{{Multiple issues | |||
|cleanup = February 2009 | |||
|refimprove = February 2009 | |||
}} | |||
In mathematics, a collection of ''n'' [[function (mathematics)|function]]s ''f''<sub>1</sub>, ''f''<sub>2</sub>, ..., ''f''<sub>''n''</sub> is '''unisolvent''' on [[domain (mathematics)|domain]] Ω if the [[vector (mathematics)|vector]]s | |||
: <math>\begin{bmatrix}f_1(x_1) \\ f_1(x_2) \\ \vdots \\ f_1(x_n)\end{bmatrix}, \begin{bmatrix}f_2(x_1) \\ f_2(x_2) \\ \vdots \\ f_2(x_n)\end{bmatrix}, \dots, \begin{bmatrix}f_n(x_1) \\ f_n(x_2) \\ \vdots \\ f_n(x_n)\end{bmatrix}</math> | |||
are [[linearly independent]] for any choice of ''n'' distinct points ''x''<sub>1</sub>, ''x''<sub>2</sub> ... ''x''<sub>''n''</sub> in Ω. Equivalently, the collection is unisolvent if the [[matrix (mathematics)|matrix]] ''F'' with entries ''f''<sub>''i''</sub>(''x''<sub>''j''</sub>) has nonzero [[determinant]]: det(''F'') ≠ 0 for any choice of distinct ''x''<sub>''j''</sub>'s in Ω. | |||
Unisolvent systems of functions are widely used in [[interpolation]] since they guarantee a unique solution to the [[interpolation problem]]. [[Polynomial]]s are unisolvent by the [[unisolvence theorem]] | |||
Examples: | |||
* 1, ''x'', ''x''<sup>2</sup> is unisolvent on any interval by the unisolvence theorem | |||
* 1, ''x''<sup>2</sup> is unisolvent on [0, 1], but not unisolvent on [−1, 1] | |||
* 1, cos(''x''), cos(2''x''), ..., cos(''nx''), sin(''x''), sin(2''x''), ..., sin(''nx'') is unisolvent on [−''π'', ''π''] | |||
Systems of unisolvent functions are much more common in 1 dimension than in higher dimensions. In dimension ''d'' = 2 and higher (Ω ⊂ '''R'''<sup>''d''</sup>), the functions ''f''<sub>1</sub>, ''f''<sub>2</sub>, ..., ''f''<sub>''n''</sub> cannot be unisolvent on Ω if there exists a single open set on which they are all continuous. To see this, consider moving points ''x''<sub>1</sub> and ''x''<sub>2</sub> along continuous paths in the open set until they have switched positions, such that ''x''<sub>1</sub> and ''x''<sub>2</sub> never intersect each other or any of the other ''x''<sub>''i''</sub>. The determinant of the resulting system (with ''x''<sub>1</sub> and ''x''<sub>2</sub> swapped) is the negative of the determinant of the initial system. Since the functions ''f''<sub>''i''</sub> are continuous, the [[intermediate value theorem]] implies that some intermediate configuration has determinant zero, hence the functions cannot be unisolvent. | |||
==References== | |||
* [[Philip J. Davis]]: ''Interpolation and Approximation'' pp. 31–32 | |||
[[Category:Interpolation]] | |||
[[Category:Numerical analysis]] | |||
[[Category:Approximation theory]] | |||
Latest revision as of 18:56, 6 March 2013
In mathematics, a collection of n functions f1, f2, ..., fn is unisolvent on domain Ω if the vectors
are linearly independent for any choice of n distinct points x1, x2 ... xn in Ω. Equivalently, the collection is unisolvent if the matrix F with entries fi(xj) has nonzero determinant: det(F) ≠ 0 for any choice of distinct xj's in Ω.
Unisolvent systems of functions are widely used in interpolation since they guarantee a unique solution to the interpolation problem. Polynomials are unisolvent by the unisolvence theorem
Examples:
- 1, x, x2 is unisolvent on any interval by the unisolvence theorem
- 1, x2 is unisolvent on [0, 1], but not unisolvent on [−1, 1]
- 1, cos(x), cos(2x), ..., cos(nx), sin(x), sin(2x), ..., sin(nx) is unisolvent on [−π, π]
Systems of unisolvent functions are much more common in 1 dimension than in higher dimensions. In dimension d = 2 and higher (Ω ⊂ Rd), the functions f1, f2, ..., fn cannot be unisolvent on Ω if there exists a single open set on which they are all continuous. To see this, consider moving points x1 and x2 along continuous paths in the open set until they have switched positions, such that x1 and x2 never intersect each other or any of the other xi. The determinant of the resulting system (with x1 and x2 swapped) is the negative of the determinant of the initial system. Since the functions fi are continuous, the intermediate value theorem implies that some intermediate configuration has determinant zero, hence the functions cannot be unisolvent.
References
- Philip J. Davis: Interpolation and Approximation pp. 31–32