A Maple Package for Parametric Matrix Computations

Transcription

1 A Maple Package for Parametric Matrix Computations Robert M. Corless, Marc Moreno Maza and Steven E. Thornton Department of Applied Mathematics, Western University Ontario Research Centre for Computer Algebra SONAD 2015 London, Canada

2 Outline 1. Motivation 2. Regular Chains 3. Background 4. Rank 5. Zigzag Form 6. Frobenius Form 7. Future Work RMC, MMM, SET ParametricMatrixTools SONAD / 47

3 Parametric Matrix Computations in CAS Computer algebra systems are currently unable to handle case discussion for parametric matrices. RMC, MMM, SET ParametricMatrixTools SONAD / 47

4 Jordan Canonical Form: Maple, Mathematica, Sage Jordan canonical form of: α α with α C RMC, MMM, SET ParametricMatrixTools SONAD / 47

5 Jordan Canonical Form: Maple, Mathematica, Sage Jordan canonical form of: α α with α C Maple 2 α α 3 RMC, MMM, SET ParametricMatrixTools SONAD / 47

6 Jordan Canonical Form: Maple, Mathematica, Sage Jordan canonical form of: α α with α C Maple Mathematica 2 α α α α RMC, MMM, SET ParametricMatrixTools SONAD / 47

7 Jordan Canonical Form: Maple, Mathematica, Sage Jordan canonical form of: α α with α C Maple Mathematica Sage 2 α α α α? RMC, MMM, SET ParametricMatrixTools SONAD / 47

8 Jordan Canonical Form: Maple, Mathematica, Sage Jordan canonical form of: α α α = RMC, MMM, SET ParametricMatrixTools SONAD / 47

9 Jordan Canonical Form: Maple, Mathematica, Sage Jordan canonical form of: α α α = Maple Mathematica Sage RMC, MMM, SET ParametricMatrixTools SONAD / 47

10 Regular Chains RMC, MMM, SET ParametricMatrixTools SONAD / 47

11 Notation Let K be an algebraically closed or real closed field Let x 1 < < x n be n 1 ordered variables K[x 1,..., x n ] is the ring of polynomials in x 1,..., x n For non-constant p K[x 1,..., x n ], mvar(p) denotes the greatest variable in p Leading coefficient of p w.r.t mvar(p) is called the initial denoted init(p) RMC, MMM, SET ParametricMatrixTools SONAD / 47

12 Notation Let K be an algebraically closed or real closed field Let x 1 < < x n be n 1 ordered variables K[x 1,..., x n ] is the ring of polynomials in x 1,..., x n For non-constant p K[x 1,..., x n ], mvar(p) denotes the greatest variable in p Leading coefficient of p w.r.t mvar(p) is called the initial denoted init(p) RMC, MMM, SET ParametricMatrixTools SONAD / 47

13 Notation Let K be an algebraically closed or real closed field Let x 1 < < x n be n 1 ordered variables K[x 1,..., x n ] is the ring of polynomials in x 1,..., x n For non-constant p K[x 1,..., x n ], mvar(p) denotes the greatest variable in p Leading coefficient of p w.r.t mvar(p) is called the initial denoted init(p) RMC, MMM, SET ParametricMatrixTools SONAD / 47

14 Notation Let K be an algebraically closed or real closed field Let x 1 < < x n be n 1 ordered variables K[x 1,..., x n ] is the ring of polynomials in x 1,..., x n For non-constant p K[x 1,..., x n ], mvar(p) denotes the greatest variable in p Leading coefficient of p w.r.t mvar(p) is called the initial denoted init(p) RMC, MMM, SET ParametricMatrixTools SONAD / 47

15 Notation Let K be an algebraically closed or real closed field Let x 1 < < x n be n 1 ordered variables K[x 1,..., x n ] is the ring of polynomials in x 1,..., x n For non-constant p K[x 1,..., x n ], mvar(p) denotes the greatest variable in p Leading coefficient of p w.r.t mvar(p) is called the initial denoted init(p) RMC, MMM, SET ParametricMatrixTools SONAD / 47

16 Regular Chain Definition (Triangular Set) A set T of non-constant polynomials in K[x 1,..., x n ] is called a triangular set if for all p, q T with p q, mvar(p) mvar(q). Definition (Saturated Ideal) The saturated ideal sat(t ) of T is the ideal T : h T, where h T is the product of the initials of the polynomials in T Definition (Regular Chain) T is a regular chain if T = or T = T {t} for t T with mvar(t) maximum such that T is a regular chain init(t) is regular modulo sat(t ) RMC, MMM, SET ParametricMatrixTools SONAD / 47

17 Regular Chain Definition (Triangular Set) A set T of non-constant polynomials in K[x 1,..., x n ] is called a triangular set if for all p, q T with p q, mvar(p) mvar(q). Definition (Saturated Ideal) The saturated ideal sat(t ) of T is the ideal T : h T, where h T is the product of the initials of the polynomials in T Definition (Regular Chain) T is a regular chain if T = or T = T {t} for t T with mvar(t) maximum such that T is a regular chain init(t) is regular modulo sat(t ) RMC, MMM, SET ParametricMatrixTools SONAD / 47

18 Regular Chain Definition (Triangular Set) A set T of non-constant polynomials in K[x 1,..., x n ] is called a triangular set if for all p, q T with p q, mvar(p) mvar(q). Definition (Saturated Ideal) The saturated ideal sat(t ) of T is the ideal T : h T, where h T is the product of the initials of the polynomials in T Definition (Regular Chain) T is a regular chain if T = or T = T {t} for t T with mvar(t) maximum such that T is a regular chain init(t) is regular modulo sat(t ) RMC, MMM, SET ParametricMatrixTools SONAD / 47

19 Constructible Set Definition (Constructible Set) A constructible set is the disjunction of systems of polynomial equations and inequations. Where the systems of equations are conjunctions of constraints. RMC, MMM, SET ParametricMatrixTools SONAD / 47

20 Semi-algebraic Set Definition (Semi-algebraic System) A semi-algebraic system of K[x 1,..., x n ] is any polynomial system S of the form f 1 = = f a = 0 g 0 p 1 > 0,..., p b > 0 q 1 0,..., q c 0 for f 1,..., f a, g, p 1,..., p b, q 1,..., q c K[x 1,..., x n ]. Definition (Semi-algebraic Set) A semi-algebraic set S of K[x 1,..., x n ] is the solution set of a semi-algebraic system. RMC, MMM, SET ParametricMatrixTools SONAD / 47

21 Semi-algebraic Set Definition (Semi-algebraic System) A semi-algebraic system of K[x 1,..., x n ] is any polynomial system S of the form f 1 = = f a = 0 g 0 p 1 > 0,..., p b > 0 q 1 0,..., q c 0 for f 1,..., f a, g, p 1,..., p b, q 1,..., q c K[x 1,..., x n ]. Definition (Semi-algebraic Set) A semi-algebraic set S of K[x 1,..., x n ] is the solution set of a semi-algebraic system. RMC, MMM, SET ParametricMatrixTools SONAD / 47

22 Example Example RealTriangularize applied to sofa and cylinder x 2 + y 3 + z 5 = x 4 + z 2 1 = 0 RMC, MMM, SET ParametricMatrixTools SONAD / 47

23 Example Example RealTriangularize applied to sofa and cylinder x 2 + y 3 + z 5 = x 4 + z 2 1 = 0 RMC, MMM, SET ParametricMatrixTools SONAD / 47

24 Example RMC, MMM, SET ParametricMatrixTools SONAD / 47

25 Outline 1. Motivation 2. Regular Chains 3. Background 4. Rank 5. Zigzag Form 6. Frobenius Form 7. Future Work RMC, MMM, SET ParametricMatrixTools SONAD / 47

26 Notation Parameters α = α 1,..., α s K is an algebraically closed or real closed field K[α] is the ring of polynomials in the parameters K(α) is the quotient field of K[α] S is a constructible (resp. semi-algebraic) set of K s RMC, MMM, SET ParametricMatrixTools SONAD / 47

27 Notation Parameters α = α 1,..., α s K is an algebraically closed or real closed field K[α] is the ring of polynomials in the parameters K(α) is the quotient field of K[α] S is a constructible (resp. semi-algebraic) set of K s RMC, MMM, SET ParametricMatrixTools SONAD / 47

28 Notation Parameters α = α 1,..., α s K is an algebraically closed or real closed field K[α] is the ring of polynomials in the parameters K(α) is the quotient field of K[α] S is a constructible (resp. semi-algebraic) set of K s RMC, MMM, SET ParametricMatrixTools SONAD / 47

29 Notation Parameters α = α 1,..., α s K is an algebraically closed or real closed field K[α] is the ring of polynomials in the parameters K(α) is the quotient field of K[α] S is a constructible (resp. semi-algebraic) set of K s RMC, MMM, SET ParametricMatrixTools SONAD / 47

30 Notation Parameters α = α 1,..., α s K is an algebraically closed or real closed field K[α] is the ring of polynomials in the parameters K(α) is the quotient field of K[α] S is a constructible (resp. semi-algebraic) set of K s RMC, MMM, SET ParametricMatrixTools SONAD / 47

31 Proposition 1,2 For two constructible (resp. semi-algebraic) sets S 1, S 2 K s one can compute a triangular decomposition of their intersection S 1 S 2 union S 1 S 2 set theoretic difference S 1 \ S 2 1 C. Chen, J. H. Davenport, J. P. May, M. Moreno Maza, B. Xia, and R. Xiao. Triangular decomposition of semi-algebraic systems. J. Symb. Comput., 49:326, C. Chen, O. Golubitsky, F. Lemaire, M. Moreno Maza, W. Pan. Comprehensive triangular decomposition. In Proc. of CASC07. Volume 4770 of Lecture Notes in Computer Science. (2007) RMC, MMM, SET ParametricMatrixTools SONAD / 47

32 Proposition 1,2 For two constructible (resp. semi-algebraic) sets S 1, S 2 K s one can compute a triangular decomposition of their intersection S 1 S 2 union S 1 S 2 set theoretic difference S 1 \ S 2 1 C. Chen, J. H. Davenport, J. P. May, M. Moreno Maza, B. Xia, and R. Xiao. Triangular decomposition of semi-algebraic systems. J. Symb. Comput., 49:326, C. Chen, O. Golubitsky, F. Lemaire, M. Moreno Maza, W. Pan. Comprehensive triangular decomposition. In Proc. of CASC07. Volume 4770 of Lecture Notes in Computer Science. (2007) RMC, MMM, SET ParametricMatrixTools SONAD / 47

33 Proposition 1,2 For two constructible (resp. semi-algebraic) sets S 1, S 2 K s one can compute a triangular decomposition of their intersection S 1 S 2 union S 1 S 2 set theoretic difference S 1 \ S 2 1 C. Chen, J. H. Davenport, J. P. May, M. Moreno Maza, B. Xia, and R. Xiao. Triangular decomposition of semi-algebraic systems. J. Symb. Comput., 49:326, C. Chen, O. Golubitsky, F. Lemaire, M. Moreno Maza, W. Pan. Comprehensive triangular decomposition. In Proc. of CASC07. Volume 4770 of Lecture Notes in Computer Science. (2007) RMC, MMM, SET ParametricMatrixTools SONAD / 47

34 Proposition 1,2 For two constructible (resp. semi-algebraic) sets S 1, S 2 K s one can compute a triangular decomposition of their intersection S 1 S 2 union S 1 S 2 set theoretic difference S 1 \ S 2 1 C. Chen, J. H. Davenport, J. P. May, M. Moreno Maza, B. Xia, and R. Xiao. Triangular decomposition of semi-algebraic systems. J. Symb. Comput., 49:326, C. Chen, O. Golubitsky, F. Lemaire, M. Moreno Maza, W. Pan. Comprehensive triangular decomposition. In Proc. of CASC07. Volume 4770 of Lecture Notes in Computer Science. (2007) RMC, MMM, SET ParametricMatrixTools SONAD / 47

35 Remark Let S K s be a constructible (resp. semi-algebraic) set; for every f(α) K[α] a partition (S eq, S neq ) of S can be computed by S eq = S V (f(α)) S neq = S \ V (f(α)) = S \ S eq. f(α) is zero everywhere on S eq f(α) is nonzero everywhere on S neq RMC, MMM, SET ParametricMatrixTools SONAD / 47

36 Remark Let S K s be a constructible (resp. semi-algebraic) set; for every f(α) K[α] a partition (S eq, S neq ) of S can be computed by S eq = S V (f(α)) S neq = S \ V (f(α)) = S \ S eq. f(α) is zero everywhere on S eq f(α) is nonzero everywhere on S neq RMC, MMM, SET ParametricMatrixTools SONAD / 47

37 Remark Let S K s be a constructible (resp. semi-algebraic) set; for every f(α) K[α] a partition (S eq, S neq ) of S can be computed by S eq = S V (f(α)) S neq = S \ V (f(α)) = S \ S eq. f(α) is zero everywhere on S eq f(α) is nonzero everywhere on S neq RMC, MMM, SET ParametricMatrixTools SONAD / 47

38 Remark Let S K s be a constructible (resp. semi-algebraic) set; for every f(α) K[α] a partition (S eq, S neq ) of S can be computed by S eq = S V (f(α)) S neq = S \ V (f(α)) = S \ S eq. f(α) is zero everywhere on S eq f(α) is nonzero everywhere on S neq RMC, MMM, SET ParametricMatrixTools SONAD / 47

39 Parametric Matrix Definition A m n parametric matrix takes the form: f 1,1 (α) f 1,2 (α) f 1,n (α) f A(α) = 2,1 (α) f 2,2 (α) f 2,n (α) f m,1 (α) f m,2 (α) f m,n (α) for f i,j K(α) 1 i m, 1 j n. Require constructible (resp. semi-algebraic) set S K s such that denominator of every entry of A(α) is nonzero everywhere on S. RMC, MMM, SET ParametricMatrixTools SONAD / 47

40 Parametric Matrix Definition A m n parametric matrix takes the form: f 1,1 (α) f 1,2 (α) f 1,n (α) f A(α) = 2,1 (α) f 2,2 (α) f 2,n (α) f m,1 (α) f m,2 (α) f m,n (α) for f i,j K(α) 1 i m, 1 j n. Require constructible (resp. semi-algebraic) set S K s such that denominator of every entry of A(α) is nonzero everywhere on S. RMC, MMM, SET ParametricMatrixTools SONAD / 47

41 Outline 1. Motivation 2. Regular Chains 3. Background 4. Rank 5. Zigzag Form 6. Frobenius Form 7. Future Work RMC, MMM, SET ParametricMatrixTools SONAD / 47

42 Parametric Rank Algorithm (Complex Case) Input: Parametric matrix A(α) Constraints on the parameter values given in a set S by a polynomial system of the form f 1 (α) = = f a (α) = 0, g(α) 0 Output: A list with elements of the form [r i, S i ] where S i gives conditions on α such that the rank of A(α) is r i Si s form a partition of S S i = S, S i S j = i j i RMC, MMM, SET ParametricMatrixTools SONAD / 47

43 Parametric Rank Algorithm (Complex Case) Input: Parametric matrix A(α) Constraints on the parameter values given in a set S by a polynomial system of the form f 1 (α) = = f a (α) = 0, g(α) 0 Output: A list with elements of the form [r i, S i ] where S i gives conditions on α such that the rank of A(α) is r i Si s form a partition of S S i = S, S i S j = i j i RMC, MMM, SET ParametricMatrixTools SONAD / 47

44 Parametric Rank Algorithm (Complex Case) Step 1: Define polynomial system S as union of equations of A(α)X = 0 and S Step 2: Let T := Triangularize(S, K[α 1 < < α s < x 1 < < x n ]) Step 3: For 0 r n, let C r be the constructible set of K s given by all regular systems [T j K[α 1 < < α s ], h j ] such that [T j, h j ] T and the number of polynomials of T j of positive degree in (at least) one of the variables x 1 < < x n is exactly n r. Step 4: For r := n down to 1 do Step 5: Project out the x i variables C r := Difference(C r, C r 1 C 0 ) RMC, MMM, SET ParametricMatrixTools SONAD / 47

45 Parametric Rank Algorithm (Complex Case) Step 1: Define polynomial system S as union of equations of A(α)X = 0 and S Step 2: Let T := Triangularize(S, K[α 1 < < α s < x 1 < < x n ]) Step 3: For 0 r n, let C r be the constructible set of K s given by all regular systems [T j K[α 1 < < α s ], h j ] such that [T j, h j ] T and the number of polynomials of T j of positive degree in (at least) one of the variables x 1 < < x n is exactly n r. Step 4: For r := n down to 1 do Step 5: Project out the x i variables C r := Difference(C r, C r 1 C 0 ) RMC, MMM, SET ParametricMatrixTools SONAD / 47

46 Parametric Rank Algorithm (Complex Case) Step 1: Define polynomial system S as union of equations of A(α)X = 0 and S Step 2: Let T := Triangularize(S, K[α 1 < < α s < x 1 < < x n ]) Step 3: For 0 r n, let C r be the constructible set of K s given by all regular systems [T j K[α 1 < < α s ], h j ] such that [T j, h j ] T and the number of polynomials of T j of positive degree in (at least) one of the variables x 1 < < x n is exactly n r. Step 4: For r := n down to 1 do Step 5: Project out the x i variables C r := Difference(C r, C r 1 C 0 ) RMC, MMM, SET ParametricMatrixTools SONAD / 47

47 Parametric Rank Algorithm (Complex Case) Step 1: Define polynomial system S as union of equations of A(α)X = 0 and S Step 2: Let T := Triangularize(S, K[α 1 < < α s < x 1 < < x n ]) Step 3: For 0 r n, let C r be the constructible set of K s given by all regular systems [T j K[α 1 < < α s ], h j ] such that [T j, h j ] T and the number of polynomials of T j of positive degree in (at least) one of the variables x 1 < < x n is exactly n r. Step 4: For r := n down to 1 do Step 5: Project out the x i variables C r := Difference(C r, C r 1 C 0 ) RMC, MMM, SET ParametricMatrixTools SONAD / 47

48 Parametric Rank Algorithm (Complex Case) Step 1: Define polynomial system S as union of equations of A(α)X = 0 and S Step 2: Let T := Triangularize(S, K[α 1 < < α s < x 1 < < x n ]) Step 3: For 0 r n, let C r be the constructible set of K s given by all regular systems [T j K[α 1 < < α s ], h j ] such that [T j, h j ] T and the number of polynomials of T j of positive degree in (at least) one of the variables x 1 < < x n is exactly n r. Step 4: For r := n down to 1 do Step 5: Project out the x i variables C r := Difference(C r, C r 1 C 0 ) RMC, MMM, SET ParametricMatrixTools SONAD / 47

49 Example 3 over C Example Consider Eẍ = A 1 ẋ + A 2 x + Bu E = A 1 = B = A 2 = λ 3 λ λ 3 λ + µ λ + µ λ + 3 µ λ, µ C 3 M. I. García-Planas and J. Clotet. Analyzing the set of uncontrollable second order generalized linear systems. International Journal of Applied Mathematics and Informatics, RMC, MMM, SET ParametricMatrixTools SONAD / 47

50 Example 3 over C Example (cont.) What is the rank of: E B A 1 E B A 2 A 1 E B C = 0 A 2 A 1 E B A 2 A B A B 3 M. I. García-Planas and J. Clotet. Analyzing the set of uncontrollable second order generalized linear systems. International Journal of Applied Mathematics and Informatics, RMC, MMM, SET ParametricMatrixTools SONAD / 47

51 Example 3 over C Example (cont.) 18 if λ 0, µ 1/2 17 if λ 0, µ = 1/2 Rank(C) = 16 if λ = 0, µ 1 15 if λ = 0, µ = 1 3 M. I. García-Planas and J. Clotet. Analyzing the set of uncontrollable second order generalized linear systems. International Journal of Applied Mathematics and Informatics, RMC, MMM, SET ParametricMatrixTools SONAD / 47

52 Zigzag Form of Parametric Matrices Zigzag Form RMC, MMM, SET ParametricMatrixTools SONAD / 47

53 Parametric Polynomial Define a polynomial in x with parameters α as f(x; α) = f 0 (α) + f 1 (α)x + + f r 1 (α)x r 1 + x r with f 0 (α),..., f r 1 (α) K(α). Require constructible (resp. semi-algebraic) set S K k such that denominator of every coefficient is nonzero everywhere on S. RMC, MMM, SET ParametricMatrixTools SONAD / 47

54 Parametric Polynomial Define a polynomial in x with parameters α as f(x; α) = f 0 (α) + f 1 (α)x + + f r 1 (α)x r 1 + x r with f 0 (α),..., f r 1 (α) K(α). Require constructible (resp. semi-algebraic) set S K k such that denominator of every coefficient is nonzero everywhere on S. RMC, MMM, SET ParametricMatrixTools SONAD / 47

55 Companion Matrix Frobenius companion matrix in x of a parametric polynomial f(x; α) = f 0 (α) + f 1 (α)x + + f r 1 (α)x r 1 + x r takes the form C f(x;α) = 0 0 f 0 (α) fr 2 (α) 1 f r 1 (α) RMC, MMM, SET ParametricMatrixTools SONAD / 47

56 Off-diagonal Block Let b {0, 1}. Define B b = b RMC, MMM, SET ParametricMatrixTools SONAD / 47

57 Zigzag Matrix 4 C c1(x;α) B b1 Cc T 2(x;α) B b2 C c3(x;α) B b3 C T c 4(x;α)... C T c d 2 (x;α) B bd 2 C cd 1 (x;α) B bd 1 C T c d (x;α) for d even 4 A. Storjohann. An O(n 3 ) algorithm for the Frobenius normal form. In Proceedings of ISSAC 1998, pages ACM, RMC, MMM, SET ParametricMatrixTools SONAD / 47

58 Theorem Theorem For every matrix A(α) K n n [α], there exists a partition (S 1,..., S N ) of the input constructible (resp. semi-algebraic) set S such that for each S i, there exists a matrix Z i (α) A(α) in Zigzag form where the denominator of the entries of Z i (α) are all nonzero everywhere on S i. RMC, MMM, SET ParametricMatrixTools SONAD / 47

59 Algorithm The algorithm extends the work of Storjohann 4 on the computation of the Zigzag form. The algorithm only uses similarity transformations to obtain the Zigzag form. Stages 1 and 3 of Storjohann s algorithm 4 are modified such that computation splits when searching for pivots. 4 A. Storjohann. An O(n 3 ) algorithm for the Frobenius normal form. In Proceedings of ISSAC 1998, pages ACM, RMC, MMM, SET ParametricMatrixTools SONAD / 47

60 Algorithm The algorithm extends the work of Storjohann 4 on the computation of the Zigzag form. The algorithm only uses similarity transformations to obtain the Zigzag form. Stages 1 and 3 of Storjohann s algorithm 4 are modified such that computation splits when searching for pivots. 4 A. Storjohann. An O(n 3 ) algorithm for the Frobenius normal form. In Proceedings of ISSAC 1998, pages ACM, RMC, MMM, SET ParametricMatrixTools SONAD / 47

61 Algorithm The algorithm extends the work of Storjohann 4 on the computation of the Zigzag form. The algorithm only uses similarity transformations to obtain the Zigzag form. Stages 1 and 3 of Storjohann s algorithm 4 are modified such that computation splits when searching for pivots. 4 A. Storjohann. An O(n 3 ) algorithm for the Frobenius normal form. In Proceedings of ISSAC 1998, pages ACM, RMC, MMM, SET ParametricMatrixTools SONAD / 47

62 Semi-Algebraic Example over R Example A(a, b) = a b with a 0, b > 0, a b RMC, MMM, SET ParametricMatrixTools SONAD / 47

63 Semi-Algebraic Example over R Example A(a, b) = a b with a 0, b > 0, a b Z 1 (a, b) = a b Z 2 (a, b) = b if a > 0, b > 0, a b if a = 0, b > 0 RMC, MMM, SET ParametricMatrixTools SONAD / 47

64 Outline 1. Motivation 2. Regular Chains 3. Background 4. Rank 5. Zigzag Form 6. Frobenius Form 7. Future Work RMC, MMM, SET ParametricMatrixTools SONAD / 47

65 What is a Frobenius Matrix A matrix in Frobenius Normal Form has the following structure: F = C f1 C f2... C fn where f n f n 1 f 1. RMC, MMM, SET ParametricMatrixTools SONAD / 47

66 What is a Frobenius Matrix A matrix in Frobenius Normal Form has the following structure: F = C f1 C f2... C fn where f n f n 1 f 1. f 1 is the minimal polynomial of F. RMC, MMM, SET ParametricMatrixTools SONAD / 47

67 Theorem Theorem For every matrix A(α) K n n [α], there exists a partition (S 1,..., S N ) of the input constructible (resp. semi-algebraic) set S such that for each S i, there exists a matrix Z i (α) F i (α) A(α) in Zigzag Frobenius form where the denominator of the entries of Z i (α) F i (α) are all nonzero everywhere on S i. RMC, MMM, SET ParametricMatrixTools SONAD / 47

68 Gcd Algorithm (Complex) Input: Two polynomials p(x; α), q(x; α) with parameters α Constraints on the parameter values given in a set S by a polynomial system of the form f 1 (α) = = f a (α) = 0, g(α) 0 Output: A list with elements of the form [g i, S i ] where S i gives conditions on α such that the gcd of p and q is g i Si s form a partition of S S i = S, S i S j = i j i RMC, MMM, SET ParametricMatrixTools SONAD / 47

69 Gcd Algorithm (Complex) Input: Two polynomials p(x; α), q(x; α) with parameters α Constraints on the parameter values given in a set S by a polynomial system of the form f 1 (α) = = f a (α) = 0, g(α) 0 Output: A list with elements of the form [g i, S i ] where S i gives conditions on α such that the gcd of p and q is g i Si s form a partition of S S i = S, S i S j = i j i RMC, MMM, SET ParametricMatrixTools SONAD / 47

70 Gcd Algorithm (Complex) Input: p, q CS R Algorithm: src := SubresultantChain(p, q, x) r := resultant(p, q, x) DS := CS V(r) for each rs DS - es := z(rs) V(init(p)) V(init(q)) - Recursive call with tail(p), tail(q), es - es := rs\es - Proceed bottom up in src and get first subresultant who s initial does not vanish on es, it is the gcd RMC, MMM, SET ParametricMatrixTools SONAD / 47

71 Gcd Example Example Consider polynomials p and q in x p(x; α) = (x 1) (x + α) q(x; α) = α x α is a complex valued parameter. RMC, MMM, SET ParametricMatrixTools SONAD / 47

72 Gcd Example Example Consider polynomials p and q in x p(x; α) = (x 1) (x + α) q(x; α) = α x α is a complex valued parameter. ( α 2 + α ) x + α if α = 0 gcd(p, q) = x 1 if α + 2 = 0 1 otherwise RMC, MMM, SET ParametricMatrixTools SONAD / 47

73 Smith Form A matrix in Smith normal form has the following structure: f 1... where f 1 f 2 f n. f n RMC, MMM, SET ParametricMatrixTools SONAD / 47

74 Smith and Frobenius Form Theorem For every square matrix A Smith form of xi A is diag(1,..., 1, f 1,..., f n ) RMC, MMM, SET ParametricMatrixTools SONAD / 47

75 Smith and Frobenius Form Theorem For every square matrix A Smith form of xi A is diag(1,..., 1, f 1,..., f n ) Frobenius form of A is given by diag(c fn,..., C f1 ) RMC, MMM, SET ParametricMatrixTools SONAD / 47

76 Computing the Smith Form Algorithm relies on gcd computations RMC, MMM, SET ParametricMatrixTools SONAD / 47

77 Computing the Smith Form Algorithm relies on gcd computations Computing xi A gives polynomials for Frobenius form RMC, MMM, SET ParametricMatrixTools SONAD / 47

78 Computing the Smith Form Algorithm relies on gcd computations Computing xi A gives polynomials for Frobenius form Faster to compute xi Z for Z in zigzag form RMC, MMM, SET ParametricMatrixTools SONAD / 47

79 Example Example A(α) = 1 α 1 0 1/2 α 2 1/2 2 3 α with α C RMC, MMM, SET ParametricMatrixTools SONAD / 47

80 Example Example A(α) = 1 α 1 0 1/2 α 2 1/2 2 3 α with α C Z 1 (α) = α 1 0 4(α 1) if α α Z 2 (α) = if α + 3 = 0 3 RMC, MMM, SET ParametricMatrixTools SONAD / 47

81 Example Example Z 1 (α) = α 1 0 4(α 1) 0 1 α 4 if α RMC, MMM, SET ParametricMatrixTools SONAD / 47

82 Example Example Z 1 (α) = α 1 0 4(α 1) 0 1 α 4 if α Smith Form of xi Z 1 (α) S 1 (α) = x 3 (α 4) x 2 (4 α 4) x 4 α RMC, MMM, SET ParametricMatrixTools SONAD / 47

83 Example Example Z 1 (α) = α 1 0 4(α 1) 0 1 α 4 if α Smith Form of xi Z 1 (α) S 1 (α) = x 3 (α 4) x 2 (4 α 4) x 4 α Frobenius Form F 1 (α) = α (α 1) 0 1 α 4 RMC, MMM, SET ParametricMatrixTools SONAD / 47

84 Example Example Z 2 (α) = if α + 3 = 0 RMC, MMM, SET ParametricMatrixTools SONAD / 47

85 Example Example Z 2 (α) = if α + 3 = 0 Smith Form of xi Z 2 (α) S 2 (α) = x 3 (α 4) x 2 (4 α 4) x 3 α + 3 RMC, MMM, SET ParametricMatrixTools SONAD / 47

86 Example Example Z 2 (α) = if α + 3 = 0 Smith Form of xi Z 2 (α) S 2 (α) = x 3 (α 4) x 2 (4 α 4) x 3 α + 3 Frobenius Form F 2 (α) = α (α 1) 0 1 α 4 RMC, MMM, SET ParametricMatrixTools SONAD / 47

87 Example Example A(α) = 1 α 1 0 1/2 α 2 1/2 2 3 α with α C F 1 (α) = F 2 (α) = α (α 1) 0 1 α α (α 1) 0 1 α 4 if α if α + 3 = 0 RMC, MMM, SET ParametricMatrixTools SONAD / 47

88 Outline 1. Motivation 2. Regular Chains 3. Background 4. Rank 5. Zigzag Form 6. Frobenius Form 7. Future Work RMC, MMM, SET ParametricMatrixTools SONAD / 47

89 Future Work Research is currently being conducted to create a Maple package ParametricMatrixTools with methods to compute Rank Frobenius (Rational) Minimal polynomial Test for similarity Jordan form Weyr form 5 5 K. O Meara, J.Clark, and C. Vinsonhaler. Advanced Topics in Linear Algebra: Weaving Matrix Problems Through the Weyr Form. Oxford University Press, 2011 RMC, MMM, SET ParametricMatrixTools SONAD / 47

90 Future Work Research is currently being conducted to create a Maple package ParametricMatrixTools with methods to compute Rank Frobenius (Rational) Minimal polynomial Test for similarity Jordan form Weyr form 5 5 K. O Meara, J.Clark, and C. Vinsonhaler. Advanced Topics in Linear Algebra: Weaving Matrix Problems Through the Weyr Form. Oxford University Press, 2011 RMC, MMM, SET ParametricMatrixTools SONAD / 47

91 Future Work Research is currently being conducted to create a Maple package ParametricMatrixTools with methods to compute Rank Frobenius (Rational) Minimal polynomial Test for similarity Jordan form Weyr form 5 5 K. O Meara, J.Clark, and C. Vinsonhaler. Advanced Topics in Linear Algebra: Weaving Matrix Problems Through the Weyr Form. Oxford University Press, 2011 RMC, MMM, SET ParametricMatrixTools SONAD / 47

92 Future Work Research is currently being conducted to create a Maple package ParametricMatrixTools with methods to compute Rank Frobenius (Rational) Minimal polynomial Test for similarity Jordan form Weyr form 5 5 K. O Meara, J.Clark, and C. Vinsonhaler. Advanced Topics in Linear Algebra: Weaving Matrix Problems Through the Weyr Form. Oxford University Press, 2011 RMC, MMM, SET ParametricMatrixTools SONAD / 47

93 Future Work Research is currently being conducted to create a Maple package ParametricMatrixTools with methods to compute Rank Frobenius (Rational) Minimal polynomial Test for similarity Jordan form Weyr form 5 5 K. O Meara, J.Clark, and C. Vinsonhaler. Advanced Topics in Linear Algebra: Weaving Matrix Problems Through the Weyr Form. Oxford University Press, 2011 RMC, MMM, SET ParametricMatrixTools SONAD / 47

94 Future Work Research is currently being conducted to create a Maple package ParametricMatrixTools with methods to compute Rank Frobenius (Rational) Minimal polynomial Test for similarity Jordan form Weyr form 5 5 K. O Meara, J.Clark, and C. Vinsonhaler. Advanced Topics in Linear Algebra: Weaving Matrix Problems Through the Weyr Form. Oxford University Press, 2011 RMC, MMM, SET ParametricMatrixTools SONAD / 47

95 Future Work Research is currently being conducted to create a Maple package ParametricMatrixTools with methods to compute Rank Frobenius (Rational) Minimal polynomial Test for similarity Jordan form Weyr form 5 5 K. O Meara, J.Clark, and C. Vinsonhaler. Advanced Topics in Linear Algebra: Weaving Matrix Problems Through the Weyr Form. Oxford University Press, 2011 RMC, MMM, SET ParametricMatrixTools SONAD / 47

96 Future Work Speed improvements Minimize unnecessary splitting Parallel implementation Cost analysis Matrix factorizations RMC, MMM, SET ParametricMatrixTools SONAD / 47

97 Future Work Speed improvements Minimize unnecessary splitting Parallel implementation Cost analysis Matrix factorizations RMC, MMM, SET ParametricMatrixTools SONAD / 47

98 Future Work Speed improvements Minimize unnecessary splitting Parallel implementation Cost analysis Matrix factorizations RMC, MMM, SET ParametricMatrixTools SONAD / 47

99 Future Work Speed improvements Minimize unnecessary splitting Parallel implementation Cost analysis Matrix factorizations RMC, MMM, SET ParametricMatrixTools SONAD / 47

100 Future Work Speed improvements Minimize unnecessary splitting Parallel implementation Cost analysis Matrix factorizations RMC, MMM, SET ParametricMatrixTools SONAD / 47

101 Code & Examples All current and future code and example worksheets will be available at StevenThornton.ca/Code Information on the RegularChains package of Maple is available at RegularChains.org RMC, MMM, SET ParametricMatrixTools SONAD / 47

102 Thank You!