divided differences
play

Divided differences Tomas Sauer Lehrstuhl fr Mathematik, - PowerPoint PPT Presentation

Jul 06, 2023 •139 likes •1.99k views

Divided differences Tomas Sauer Lehrstuhl fr Mathematik, Schwerpunkt Digitale Bildverarbeitung FORWISS Universitt Passau MAIA 2013, Erice, September 26, 2013 In part joint work with J. Carnicer (Zaragoza) Tomas Sauer (Universitt Passau)

  1. Different Points of View Point set constructions Given a space P ⊂ Π find sites Ξ such that there always exists a unique p ∈ F with p ( Ξ ) = f ( Ξ ) . Answers: Chung–Yao, GPL, . . . Point set constructions Find smallest subspace P ⊂ Π such that for any Ξ with # Ξ = N there exists p ∈ P with p ( Ξ ) = f ( Ξ ) . � n + 2 � Answers: Π N − 1 , Π 2 n − 1 , = N , for s = 2. 2 Space constructions Given Ξ find P such that there always exists p ∈ P with p ( Ξ ) = f ( Ξ ) . Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 4 / 24

  2. Different Points of View Point set constructions Given a space P ⊂ Π find sites Ξ such that there always exists a unique p ∈ F with p ( Ξ ) = f ( Ξ ) . Answers: Chung–Yao, GPL, . . . Point set constructions Find smallest subspace P ⊂ Π such that for any Ξ with # Ξ = N there exists p ∈ P with p ( Ξ ) = f ( Ξ ) . � n + 2 � Answers: Π N − 1 , Π 2 n − 1 , = N , for s = 2. 2 Space constructions Given Ξ find P such that there always exists p ∈ P with p ( Ξ ) = f ( Ξ ) . Answers: Buchberger–Möller, deBoor–Ron, ideal remainders Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 4 / 24

  3. Notation Polynomials Π polynomials = finite sums 1 � p α x α . p ( x ) = | α | ≤ n deg p := n . 2 Monomials or terms of degree k , . . . , n : row vectors 3 x k : n := (( · ) α : k ≤ | α | ≤ n ) , x n = x n : n Coefficients as column vectors p k = ( p α : | α | = k ) . 4 n � x k p k . Polynomials: p ( x ) = 5 k = 0 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 5 / 24

  4. Notation Polynomials Π polynomials = finite sums 1 � p α x α . p ( x ) = | α | ≤ n deg p := n . 2 Monomials or terms of degree k , . . . , n : row vectors 3 x k : n := (( · ) α : k ≤ | α | ≤ n ) , x n = x n : n Coefficients as column vectors p k = ( p α : | α | = k ) . 4 n � x k p k . Polynomials: p ( x ) = 5 k = 0 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 5 / 24

  5. Notation Polynomials Π polynomials = finite sums 1 � p α x α . p ( x ) = | α | ≤ n Degree deg p := n . 2 Monomials or terms of degree k , . . . , n : row vectors 3 x k : n := (( · ) α : k ≤ | α | ≤ n ) , x n = x n : n Coefficients as column vectors p k = ( p α : | α | = k ) . 4 n � x k p k . Polynomials: p ( x ) = 5 k = 0 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 5 / 24

  6. Notation Polynomials Π polynomials = finite sums 1 � p α x α . p ( x ) = | α | ≤ n Total degree deg p := n . 2 Monomials or terms of degree k , . . . , n : row vectors 3 x k : n := (( · ) α : k ≤ | α | ≤ n ) , x n = x n : n Coefficients as column vectors p k = ( p α : | α | = k ) . 4 n � x k p k . Polynomials: p ( x ) = 5 k = 0 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 5 / 24

  7. Notation Polynomials Π polynomials = finite sums 1 � p α x α . p ( x ) = | α | ≤ n Total degree deg p := n . 2 Monomials or terms of degree k , . . . , n : row vectors 3 x k : n := (( · ) α : k ≤ | α | ≤ n ) , x n = x n : n Coefficients as column vectors p k = ( p α : | α | = k ) . 4 n � x k p k . Polynomials: p ( x ) = 5 k = 0 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 5 / 24

  8. Notation Polynomials Π polynomials = finite sums 1 � p α x α . p ( x ) = | α | ≤ n Total degree deg p := n . 2 Monomials or terms of degree k , . . . , n : row vectors 3 x k : n := (( · ) α : k ≤ | α | ≤ n ) , x n = x n : n Coefficients as column vectors p k = ( p α : | α | = k ) . 4 n � x k p k . Polynomials: p ( x ) = 5 k = 0 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 5 / 24

  9. Notation Polynomials Π polynomials = finite sums 1 � p α x α . p ( x ) = | α | ≤ n Total degree deg p := n . 2 Monomials or terms of degree k , . . . , n : row vectors 3 x k : n := (( · ) α : k ≤ | α | ≤ n ) , x n = x n : n Coefficients as column vectors p k = ( p α : | α | = k ) . 4 n � x k p k . Polynomials: p ( x ) = 5 k = 0 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 5 / 24

  10. Interpolation Correctness A subspace P ⊂ Π is called correct for Ξ ⊂ R s if for any f : Ξ → R there exists unique L Ξ f ∈ P such that L Ξ f ( Ξ ) = f ( Ξ ) . Correctness for Π n Vandermonde matrix � � ξ ∈ Ξ x 0 : n ( Ξ ) = ξ α : | α | ≤ n Then: L Ξ f = x 0 : n p , Fundamental “Theorem” Correctness = nonsingularity of Vandermonde matrix. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 6 / 24

  11. Interpolation Correctness A subspace P ⊂ Π is called correct for Ξ ⊂ R s if for any f : Ξ → R there exists unique L Ξ f ∈ P such that L Ξ f ( Ξ ) = f ( Ξ ) . Correctness for Π n Vandermonde matrix � � ξ ∈ Ξ x 0 : n ( Ξ ) = ξ α : | α | ≤ n Then: L Ξ f = x 0 : n p , x 0 : n ( Ξ ) p = f ( Ξ ) . Fundamental “Theorem” Correctness = nonsingularity of Vandermonde matrix. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 6 / 24

  12. Interpolation Correctness A subspace P ⊂ Π is called correct for Ξ ⊂ R s if for any f : Ξ → R there exists unique L Ξ f ∈ P such that L Ξ f ( Ξ ) = f ( Ξ ) . Correctness for Π n Vandermonde matrix � � ξ ∈ Ξ x 0 : n ( Ξ ) = ξ α : | α | ≤ n Then: p = x 0 : n ( Ξ ) − 1 f ( Ξ ) . L Ξ f = x 0 : n p , Fundamental “Theorem” Correctness = nonsingularity of Vandermonde matrix. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 6 / 24

  13. Interpolation Correctness A subspace P ⊂ Π is called correct for Ξ ⊂ R s if for any f : Ξ → R there exists unique L Ξ f ∈ P such that L Ξ f ( Ξ ) = f ( Ξ ) . Correctness for Π n Vandermonde matrix � � ξ ∈ Ξ x 0 : n ( Ξ ) = ξ α : | α | ≤ n Then: p = x 0 : n ( Ξ ) − 1 f ( Ξ ) . L Ξ f = x 0 : n p , Fundamental “Theorem” Correctness = nonsingularity of Vandermonde matrix. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 6 / 24

  14. Interpolation Spaces Degree of a subspace deg P = max { deg p : p ∈ P } . Degree reducing interpolation f ∈ Π deg L Ξ f ≤ deg f . ⇒ Minimal degree interpolation space Q interpolation space deg Q ≥ deg P . ⇒ Facts Degree reducing is always minimal degree. 1 None of them is unique for general Ξ . 2 Different elimination strategies for x 0 : n ( Ξ ) . 3 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 7 / 24

  15. Interpolation Spaces Degree of a subspace deg P = max { deg p : p ∈ P } . Degree reducing interpolation f ∈ Π deg L Ξ f ≤ deg f . ⇒ Minimal degree interpolation space Q interpolation space deg Q ≥ deg P . ⇒ Facts Degree reducing is always minimal degree. 1 None of them is unique for general Ξ . 2 Different elimination strategies for x 0 : n ( Ξ ) . 3 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 7 / 24

  16. Interpolation Spaces Degree of a subspace deg P = max { deg p : p ∈ P } . Degree reducing interpolation f ∈ Π deg L Ξ f ≤ deg f . ⇒ Minimal degree interpolation space Q interpolation space deg Q ≥ deg P . ⇒ Facts Degree reducing is always minimal degree. 1 None of them is unique for general Ξ . 2 Different elimination strategies for x 0 : n ( Ξ ) . 3 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 7 / 24

  17. Interpolation Spaces Degree of a subspace deg P = max { deg p : p ∈ P } . Degree reducing interpolation f ∈ Π deg L Ξ f ≤ deg f . ⇒ Minimal degree interpolation space Q interpolation space deg Q ≥ deg P . ⇒ Facts Degree reducing is always minimal degree. 1 None of them is unique for general Ξ . 2 Different elimination strategies for x 0 : n ( Ξ ) . 3 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 7 / 24

  18. Interpolation Spaces Degree of a subspace deg P = max { deg p : p ∈ P } . Degree reducing interpolation f ∈ Π deg L Ξ f ≤ deg f . ⇒ Minimal degree interpolation space Q interpolation space deg Q ≥ deg P . ⇒ Facts Degree reducing is always minimal degree. 1 None of them is unique for general Ξ . 2 Different elimination strategies for x 0 : n ( Ξ ) . 3 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 7 / 24

  19. Interpolation Spaces Degree of a subspace deg P = max { deg p : p ∈ P } . Degree reducing interpolation f ∈ Π deg L Ξ f ≤ deg f . ⇒ Minimal degree interpolation space Q interpolation space deg Q ≥ deg P . ⇒ Facts Degree reducing is always minimal degree. 1 None of them is unique for general Ξ . 2 Different elimination strategies for x 0 : n ( Ξ ) . 3 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 7 / 24

  20. Interpolation Spaces Degree of a subspace deg P = max { deg p : p ∈ P } . Degree reducing interpolation f ∈ Π deg L Ξ f ≤ deg f . ⇒ Minimal degree interpolation space Q interpolation space deg Q ≥ deg P . ⇒ Facts Degree reducing is always minimal degree. 1 None of them is unique for general Ξ . 2 Different elimination strategies for x 0 : n ( Ξ ) . 3 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 7 / 24

  21. Interpolation Spaces Degree of a subspace deg P = max { deg p : p ∈ P } . Degree reducing interpolation f ∈ Π deg L Ξ f ≤ deg f . ⇒ Minimal degree interpolation space Q interpolation space deg Q ≥ deg P . ⇒ Facts Degree reducing is always minimal degree. 1 None of them is unique for general Ξ . 2 Different elimination strategies for x 0 : n ( Ξ ) . 3 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 7 / 24

  22. The Two Faces of Polynomials The linear part Computation of interpolant: solve Vandermonde system . 1 But . . . Polynomials can be multiplied . 1 Polynomial algebra . . . 2 T. Pratchett, Jingo There’s al–gebra. That’s like sums with letters. For . . . for people whose brains aren’t clever enough for numbers, see? Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 8 / 24

  23. The Two Faces of Polynomials The linear part Computation of interpolant: solve Vandermonde system . 1 But . . . Polynomials can be multiplied . 1 Polynomial algebra . . . 2 T. Pratchett, Jingo There’s al–gebra. That’s like sums with letters. For . . . for people whose brains aren’t clever enough for numbers, see? Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 8 / 24

  24. The Two Faces of Polynomials The linear part Computation of interpolant: solve Vandermonde system . 1 But . . . Polynomials can be multiplied . 1 Polynomial algebra . . . 2 T. Pratchett, Jingo There’s al–gebra. That’s like sums with letters. For . . . for people whose brains aren’t clever enough for numbers, see? Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 8 / 24

  25. The Two Faces of Polynomials The linear part Computation of interpolant: solve Vandermonde system . 1 But . . . Polynomials can be multiplied . 1 Polynomial algebra . . . 2 T. Pratchett, Jingo There’s al–gebra. That’s like sums with letters. For . . . for people whose brains aren’t clever enough for numbers, see? Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 8 / 24

  26. The Two Faces of Polynomials The linear part Computation of interpolant: solve Vandermonde system . 1 But . . . Polynomials can be multiplied . 1 Polynomial algebra . . . 2 T. Pratchett, Jingo There’s al–gebra. That’s like sums with letters. For . . . for people whose brains aren’t clever enough for numbers, see? Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 8 / 24

  27. The Two Faces of Polynomials The linear part Computation of interpolant: solve Vandermonde system . 1 But . . . Polynomials can be multiplied . 1 Polynomial algebra . . . 2 T. Pratchett, Jingo There’s al–gebra. That’s like sums with letters. For . . . for people whose brains aren’t clever enough for numbers, see? Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 8 / 24

  28. Ideals Ideals Ideal I : 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  29. Ideals Ideals Ideal I : 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  30. Ideals Ideals Ideal I : I + I = I 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  31. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  32. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  33. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  34. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 f F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  35. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 f g f : g f ∈ Π F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  36. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3 � f g f : g f ∈ Π f ∈ F F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  37. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3     � � F � =: f g f : g f ∈ Π  .  f ∈ F F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  38. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3     � � F � =: f g f : g f ∈ Π  .  f ∈ F F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  39. Ideals Ideals Ideal I : I + I = I and I · Π = I . 1 I ( Ξ ) := { f ∈ Π : f ( Ξ ) = 0 } . 2 Ideal generated by F ⊂ Π : 3     � � F � =: f g f : g f ∈ Π  .  f ∈ F F basis of I = � F � . 4 Hilbert’s Basissatz Any polynomial ideal has a finite basis. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 9 / 24

  40. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . 3 Set 4 Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  41. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . 3 Set 4 Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  42. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . 3 Set 4 Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  43. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . 3 Set 4 Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  44. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  45. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  46. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 p ∈ P ∗ Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  47. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 p − L Ξ p : p ∈ P ∗ Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  48. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 F Ξ := { p − L Ξ p : p ∈ P ∗ } . Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  49. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 F Ξ := { p − L Ξ p : p ∈ P ∗ } . Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  50. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 F Ξ := { p − L Ξ p : p ∈ P ∗ } . Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  51. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 F Ξ := { p − L Ξ p : p ∈ P ∗ } . Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : I ( Ξ ) ∋ g Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  52. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 F Ξ := { p − L Ξ p : p ∈ P ∗ } . Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : � I ( Ξ ) ∋ g = g f f f ∈ F Ξ Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  53. From Interpolation to Bases Construction of basis P degree reducing interpolation space for Ξ . 1 P basis for P . 2 � � Set P ∗ = ( · ) j p : p ∈ P , j = 1, . . . , s . (Pre-)border basis. 3 Set 4 F Ξ := { p − L Ξ p : p ∈ P ∗ } . Then: I ( Ξ ) = � F Ξ � . 5 Even better Basis is H–basis : � I ( Ξ ) ∋ g = g f f , deg g f f ≤ deg g . f ∈ F Ξ Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 10 / 24

  54. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  55. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  56. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  57. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 f = ω p + r , L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  58. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 deg r ≤ deg ω − 1 f = ω p + r , L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  59. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 deg r ≤ deg ω − 1 = # Ξ − 1. f = ω p + r , L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  60. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 deg r ≤ deg ω − 1 = # Ξ − 1. f = ω p + r , L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  61. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 deg r ≤ deg ω − 1 = # Ξ − 1. f = ω p + r , L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  62. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 deg r ≤ deg ω − 1 = # Ξ − 1. f = ω p + r , L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  63. Implicit Interpolation The case s = 1 � ( · − ξ ) . ω = 1 ξ ∈ Ξ Division with remainder: 2 deg r ≤ deg ω − 1 = # Ξ − 1. f = ω p + r , L Ξ f := r interpolation polynomial. 3 Algebraic interpretation Principal ideal: I ( Ξ ) = � ω � . 1 Ideal + Quotient Space. 2 Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 11 / 24

  64. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  65. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  66. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  67. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  68. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  69. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  70. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  71. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Ξ Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  72. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Ξ � → I ( Ξ ) Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

  73. Implicit Interpolation II Division with remainder Division by set F : 1 � g = g f f + ν F ( g ) . f ∈ F Ideal + normal form. 2 Normal form is 3 interpolant. 1 unique if F is H–basis. 2 Constructive chain Ξ � → I ( Ξ ) � → H–basis F Remark All degree reducing interpolants can be constructed this way. Tomas Sauer (Universität Passau) Divided differences MAIA 2013, Erice 12 / 24

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Interpolation & Polynomial Approximation Divided Differences: A Brief Introduction Numerical

Interpolation & Polynomial Approximation Divided Differences: A Brief Introduction Numerical

Interpolation & Polynomial Approximation Divided Differences: A Brief Introduction Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011

474 views • 32 slides
6. Individual Differences Differences: Big Questions Are some differences changeable and

6. Individual Differences Differences: Big Questions Are some differences changeable and

6. Individual Differences Differences: Big Questions Are some differences changeable and how much? What effect does community and environment play? What can teachers do to accommodate differences? 6.1 The Nature-Nurture

742 views • 52 slides
of Israel and Judah Every kingdom divide divided against itself will be ruined, and every

of Israel and Judah Every kingdom divide divided against itself will be ruined, and every

The Divided Kingdoms of Israel and Judah Every kingdom divide divided against itself will be ruined, and every city or house household divided divided against itself will not stand. (Matt 12:25)

803 views • 5 slides
Designing for differences Dan Smith 2 Designing for Differences - Goals Be familiar with

Designing for differences Dan Smith 2 Designing for Differences - Goals Be familiar with

1 Designing for differences Dan Smith 2 Designing for Differences - Goals Be familiar with the issues that affect usability and interaction for different individual and groups of people Know the basic tools for designing interactive

525 views • 26 slides
Divided symmetrization, quasisymmetric functions and Schubert polynomials Vasu Tewari University

Divided symmetrization, quasisymmetric functions and Schubert polynomials Vasu Tewari University

Divided symmetrization, quasisymmetric functions and Schubert polynomials Vasu Tewari University of Pennsylvania (joint with Philippe Nadeau, Universit e Lyon 1) AMS Sectional Meeting Gainesville, 3rd November 2019 Postnikovs divided

601 views • 59 slides
The Problem of the Divided Majority Preference Aggregation and Uncertainty ura-Georg Grani

The Problem of the Divided Majority Preference Aggregation and Uncertainty ura-Georg Grani

The Problem of the Divided Majority Preference Aggregation and Uncertainty ura-Georg Grani c, University of Cologne, Germany <georg.granic@uni-koeln.de> The Problem of the Divided Majority p. 1 The Divided Majority The Problem

671 views • 36 slides
Differences-in-Differences Estimator: Example Card and Krueger (1994) Example : Card D. and

Differences-in-Differences Estimator: Example Card and Krueger (1994) Example : Card D. and

Differences-in-Differences Estimator: Example Card and Krueger (1994) Example : Card D. and Krueger A. (1994) Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania, American Economic Review,

1.71k views • 78 slides
Time Spent on Research with Undergraduate Students Gender Differences among Gender Differences

Time Spent on Research with Undergraduate Students Gender Differences among Gender Differences

Time Spent on Research with Undergraduate Students Gender Differences among Gender Differences among STEM Faculty Amber D. Lambert Amy K. Garver Allison BrckaLorenz Indiana University Antwione Haywood Center for Postsecondary Research

235 views • 12 slides
NJIPLA - Electronics, Telecom and Software Patent Practice Update Indirect Infringement/Divided

NJIPLA - Electronics, Telecom and Software Patent Practice Update Indirect Infringement/Divided

NJIPLA - Electronics, Telecom and Software Patent Practice Update Indirect Infringement/Divided Infringement David Rosenblatt, Assistant General Counsel Intellectual Property Thomson Reuters January 24, 2013 Topics Indirect/divided

627 views • 15 slides
M aterials can be divided into two basic categories: struc- 1 Auxetic behaviour tural or

M aterials can be divided into two basic categories: struc- 1 Auxetic behaviour tural or

A triumph of lateral thought ANDREW ALDERSON Imagine stretching elastic and seeing it get fatter rather than thinner. It may sound bizarre, but this property is what makes auxetic materials potentially so useful M aterials can be divided into

389 views • 6 slides
Scenario 1: A multipolar divided world, where global challenges are left unchecked Lonely

Scenario 1: A multipolar divided world, where global challenges are left unchecked Lonely

Scenario 1: A multipolar divided world, where global challenges are left unchecked Lonely Neigbours What does 2020 look like? New year 2020 sets off in a divided world. The power and influence of different international and multilateral

189 views • 3 slides
1 Introduction Darts is a game played on a circular board (figure 1) divided into twenty num-

1 Introduction Darts is a game played on a circular board (figure 1) divided into twenty num-

1 Introduction 1 1 Introduction Darts is a game played on a circular board (figure 1) divided into twenty num- bered segments. Each segment is divided three areas, single, double and triple. When a dart hits a segment the score is double or

228 views • 19 slides
Domain and Genre differences Martin Smrt martin@martinsmrt.com Contents Motivation

Domain and Genre differences Martin Smrt martin@martinsmrt.com Contents Motivation

Domain and Genre differences Martin Smrt martin@martinsmrt.com Contents Motivation Cross-register differences Grammatical issues Lexicographic issues Multidimensional differences Applications Approaches to Corpus Building

457 views • 26 slides
1 AD pathology is more likely to be clinically 20% more women than men die of Alzheimers 20%

1 AD pathology is more likely to be clinically 20% more women than men die of Alzheimers 20%

Disclosures Disclosures Objectives Objectives Sex Differences in Alzheimers Disease 1. Describe sex differences in the prevalence and 1. Describe sex differences in the prevalence and travel support from Abbott incidence of

357 views • 6 slides
SQL3 / SQL:2008 The SQL3 standard is big and is therefore divided into parts: DD2471 (Lecture 08)

SQL3 / SQL:2008 The SQL3 standard is big and is therefore divided into parts: DD2471 (Lecture 08)

SQL3 / SQL:2008 The SQL3 standard is big and is therefore divided into parts: DD2471 (Lecture 08) Modern database systems & their applications Spring 2012 1 / 44 SQL3 / SQL:2008 The SQL3 standard is big and is therefore divided into parts:

1.56k views • 155 slides
Should the Olympics be Divided into Mens and Womens Events? 1 ENGAGE Silent Written

Should the Olympics be Divided into Mens and Womens Events? 1 ENGAGE Silent Written

Should the Olympics be Divided into Mens and Womens Events? 1 ENGAGE Silent Written Response: Should the Olympics be divided into Mens and Womens Events? Explain your reasoning. 2 ENGAGE DISCUSS with your elbow partner:

436 views • 28 slides
Analysis of Multithreaded Algorithms Marc Moreno Maza University of Western Ontario, London,

Analysis of Multithreaded Algorithms Marc Moreno Maza University of Western Ontario, London,

Analysis of Multithreaded Algorithms Marc Moreno Maza University of Western Ontario, London, Ontario (Canada) CS4402-9535 (Moreno Maza) Analysis of Multithreaded Algorithms CS4402-9535 1 / 47 Plan Review of Complexity Notions 1

794 views • 47 slides
Sorting Should we worry about speed? Task Description We have an array of n values in any

Sorting Should we worry about speed? Task Description We have an array of n values in any

Sorting Should we worry about speed? Task Description We have an array of n values in any order We need to have the array sorted in ascending or descending order of values 2 Selection Sort Select the smallest value in the array

735 views • 8 slides
Divide & Conquer

Divide & Conquer

Divide & Conquer (

574 views • 23 slides
Informatik II Ubung 4 FS 2020 1 Program Today Survey Productive Failure Case Study 1 1

Informatik II Ubung 4 FS 2020 1 Program Today Survey Productive Failure Case Study 1 1

Informatik II Ubung 4 FS 2020 1 Program Today Survey Productive Failure Case Study 1 1 Feedback of last exercise 2 Repetition Theory 3 Analysis of programs Analysis of recurrences Divide & Conquer Sorting Algorithms 4 2 3.

760 views • 64 slides
Quantifying the sensitivity of U.S. ozone concentrations to domestic vs international emissions

Quantifying the sensitivity of U.S. ozone concentrations to domestic vs international emissions

Quantifying the sensitivity of U.S. ozone concentrations to domestic vs international emissions through coupled GEOS-Chem Adjoint and CMAQ DDM source-receptor modeling Or: The Boundary Sensitivity Project Farhan Akhtar, Barron Henderson,

659 views • 38 slides
arXiv:2006.16995v1 [math.CO] 30 Jun 2020 1. Introduction 1.1. Schubert polynomials and pipe

arXiv:2006.16995v1 [math.CO] 30 Jun 2020 1. Introduction 1.1. Schubert polynomials and pipe

SLIDE POLYNOMIALS AND SUBWORD COMPLEXES EVGENY SMIRNOV AND ANNA TUTUBALINA Abstract. Subword complexes were defined by A. Knutson and E. Miller in 2004 for describing Gr obner degenerations of matrix Schubert varieties. The facets of such a

585 views • 15 slides
Finding Line Bundle bases in Equivariant K-theory J. Frias, J. Rossi, Advised by Dr. Rebecca

Finding Line Bundle bases in Equivariant K-theory J. Frias, J. Rossi, Advised by Dr. Rebecca

Finding Line Bundle bases in Equivariant K-theory J. Frias, J. Rossi, Advised by Dr. Rebecca Goldin Flags A (complete) flag of C n is a chain of inclusions of vector subspaces { 0 } V 1 V n 1 C n , where dim V k = k .

378 views • 16 slides
UMBC A B M A L T F O U M B C I M Y O R T 1 (11/6/07) I E S R C E O V U

UMBC A B M A L T F O U M B C I M Y O R T 1 (11/6/07) I E S R C E O V U

Programmable Logic Devices Verilog Design Examples CMPE 415 Building Blocks Digital systems consist of 2 main parts: the datapath and control circuits. Datapath : stores and manipulates data and includes components such as registers, shift

348 views • 17 slides

More recommend