On the growth factor for Hadamard matrices
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions
Christos Kravvaritis joint work with Marilena Mitrouli
Importance
History Determinants Preliminaries
University of Athens Department of Mathematics
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
Householder Symposium XVII - Zeuthen 2008
Outline Introduction Gaussian Elimination Definitions Importance of this study
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
History of the problem Determinants Preliminary Results
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
Outline Introduction Gaussian Elimination Definitions Importance of this study
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
History of the problem Determinants Preliminary Results
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices
I
linear system Ax = b, A = [aij ] ∈ Rn×n
I
Gaussian Elimination (GE): a11 a12 · · · a1n a21 a22 · · · a2n A= .. .. .. −→ . . ··· .
A(n−1)
an1 an2 · · · ann a11 a12 · · · 0 a(1) · · · 22 (2) 0 0 a33 = .. .. .. . . . 0 0 0
C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History Determinants Preliminaries
Solution The proposed idea
··· ··· ··· .. . ···
a1n (1) a2n (2) a3n .. . (n−1)
ann
Pivots from the beginning Pivots from the end
Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices
I
linear system Ax = b, A = [aij ] ∈ Rn×n
I
Gaussian Elimination (GE): a11 a12 · · · a1n a21 a22 · · · a2n A= .. .. .. −→ . . ··· .
A(n−1)
an1 an2 · · · ann a11 a12 · · · 0 a(1) · · · 22 (2) 0 0 a33 = .. .. .. . . . 0 0 0
C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History Determinants Preliminaries
Solution The proposed idea
··· ··· ··· .. . ···
a1n (1) a2n (2) a3n .. . (n−1)
ann
Pivots from the beginning Pivots from the end
Numerical experiments Pivot patterns
SummaryReferences
Backward error analysis for GE −→ growth factor (k )
g(n, A) =
maxi,j,k |aij | maxi,j |aij |
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History Determinants
Theorem (Wilkinson) The computed solution xˆ to the linear system Ax = b using GE with partial pivoting satisfies
Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
(A + ∆A)xˆ = b with k∆Ak∞ ≤ cn3 g(n, A)kAk∞ u.
SummaryReferences
Backward error analysis for GE −→ growth factor (k )
g(n, A) =
maxi,j,k |aij | maxi,j |aij |
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History Determinants
Theorem (Wilkinson) The computed solution xˆ to the linear system Ax = b using GE with partial pivoting satisfies
Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
(A + ∆A)xˆ = b with k∆Ak∞ ≤ cn3 g(n, A)kAk∞ u.
SummaryReferences
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
Definition Completely pivoted (CP) matrices: no row and column exchanges are needed during GE with complete pivoting.
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
Outline Introduction Gaussian Elimination Definitions Importance of this study
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
History of the problem Determinants Preliminary Results
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices
Definition A Hadamard matrix H of order n (symb. Hn ) is a ±1 matrix satisfying
C. Kravvaritis Introduction Gaussian Elimination Definitions
HH T = H T H = nIn .
Importance
History Determinants
Important property: every two distinct rows and columns of a Hadamard matrix are orthogonal.
Preliminaries
Solution The proposed idea Pivots from the beginning
Example
Pivots from the end Numerical experiments Pivot patterns
1 1 1 1 1 −1 1 −1 H4 = 1 1 −1 −1 1 −1 −1 1
SummaryReferences
On the growth factor for Hadamard matrices
Definition A Hadamard matrix H of order n (symb. Hn ) is a ±1 matrix satisfying
C. Kravvaritis Introduction Gaussian Elimination Definitions
HH T = H T H = nIn .
Importance
History Determinants
Important property: every two distinct rows and columns of a Hadamard matrix are orthogonal.
Preliminaries
Solution The proposed idea Pivots from the beginning
Example
Pivots from the end Numerical experiments Pivot patterns
1 1 1 1 1 −1 1 −1 H4 = 1 1 −1 −1 1 −1 −1 1
SummaryReferences
On the growth factor for Hadamard matrices
Definition A Hadamard matrix H of order n (symb. Hn ) is a ±1 matrix satisfying
C. Kravvaritis Introduction Gaussian Elimination Definitions
HH T = H T H = nIn .
Importance
History Determinants
Important property: every two distinct rows and columns of a Hadamard matrix are orthogonal.
Preliminaries
Solution The proposed idea Pivots from the beginning
Example
Pivots from the end Numerical experiments Pivot patterns
1 1 1 1 1 −1 1 −1 H4 = 1 1 −1 −1 1 −1 −1 1
SummaryReferences
On the growth factor for Hadamard matrices
Example
C. Kravvaritis
H16
=
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1−1 1−1 1−1 1−1 1−1 1−1 1−1 1−1 1 1−1−1 1 1−1−1 1 1−1−1 1 1−1−1 1−1−1 1 1−1−1 1 1−1−1 1 1−1−1 1 1 1 1 1−1−1−1−1 1 1 1 1−1−1−1−1 1−1 1−1−1 1−1 1 1−1 1−1−1 1−1 1 1 1−1−1−1−1 1 1 1 1−1−1−1−1 1 1 1−1−1 1−1 1 1−1 1−1−1 1−1 1 1−1 1 1 1 1 1 1 1 1−1−1−1−1−1−1−1−1 1−1 1−1 1−1 1−1−1 1−1 1−1 1−1 1 1 1−1−1 1 1−1−1−1−1 1 1−1−1 1 1 1−1−1 1 1−1−1 1−1 1 1−1−1 1 1−1 1 1 1 1−1−1−1−1−1−1−1−1 1 1 1 1 1−1 1−1−1 1−1 1−1 1−1 1 1−1 1−1 1 1−1−1−1−1 1 1−1−1 1 1 1 1−1−1
1−1−1 1−1 1 1−1−1 1 1−1 1−1−1 1
Introduction Gaussian Elimination Definitions Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination
Definition Two matrices are said to be Hadamard equivalent or H-equivalent if one can be obtained from the other by a sequence of the operations: 1. interchange any pairs of rows and/or columns;
Definitions Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end
2. multiply any rows and/or columns through by −1.
Numerical experiments Pivot patterns
SummaryReferences
Outline Introduction Gaussian Elimination Definitions Importance of this study
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
History of the problem Determinants Preliminary Results
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices C. Kravvaritis
Why Hadamard matrices? 1. Numerous applications in various areas of modern Mathematics: I I I I I I I
Statistics-Theory of Experimental Designs Coding Theory Cryptography Combinatorics Image Processing Signal Processing Analytical Chemistry
2. Interesting properties regarding the size of the pivots appearing after application of GE
Introduction Gaussian Elimination Definitions Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices C. Kravvaritis
Why Hadamard matrices? 1. Numerous applications in various areas of modern Mathematics: I I I I I I I
Statistics-Theory of Experimental Designs Coding Theory Cryptography Combinatorics Image Processing Signal Processing Analytical Chemistry
2. Interesting properties regarding the size of the pivots appearing after application of GE
Introduction Gaussian Elimination Definitions Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices C. Kravvaritis
Why Hadamard matrices? 1. Numerous applications in various areas of modern Mathematics: I I I I I I I
Statistics-Theory of Experimental Designs Coding Theory Cryptography Combinatorics Image Processing Signal Processing Analytical Chemistry
2. Interesting properties regarding the size of the pivots appearing after application of GE
Introduction Gaussian Elimination Definitions Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
History
On the growth factor for Hadamard matrices C. Kravvaritis
I
Tornheim, 1964 g(n, H) ≥ n for a CP n × n Hadamard matrix H
Introduction Gaussian Elimination Definitions Importance
History
I
Cryer, 1968 Conjecture: g(n, A) ≤ n, with equality iff A is a Hadamard matrix
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments
I
Day & Peterson, 1988 proved the equality only for the Hadamard-Sylvester Hn Hn class H2n = Hn −Hn Conjecture: pn−3 = n/4
Pivot patterns
SummaryReferences
History
On the growth factor for Hadamard matrices C. Kravvaritis
I
Tornheim, 1964 g(n, H) ≥ n for a CP n × n Hadamard matrix H
Introduction Gaussian Elimination Definitions Importance
History
I
Cryer, 1968 Conjecture: g(n, A) ≤ n, with equality iff A is a Hadamard matrix
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments
I
Day & Peterson, 1988 proved the equality only for the Hadamard-Sylvester Hn Hn class H2n = Hn −Hn Conjecture: pn−3 = n/4
Pivot patterns
SummaryReferences
History
On the growth factor for Hadamard matrices C. Kravvaritis
I
Tornheim, 1964 g(n, H) ≥ n for a CP n × n Hadamard matrix H
Introduction Gaussian Elimination Definitions Importance
History
I
Cryer, 1968 Conjecture: g(n, A) ≤ n, with equality iff A is a Hadamard matrix
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments
I
Day & Peterson, 1988 proved the equality only for the Hadamard-Sylvester Hn Hn class H2n = Hn −Hn Conjecture: pn−3 = n/4
Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices
I
Gould, 1991 found a 13 × 13 matrix with growth 13.0205 The first part of Cryer’s conjecture is false. The second part still remains open
C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
I
Edelman & Mascarenhas, 1995 g(12, H12 ) = 12
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
I
I
Edelman & Friedman, 1998 found the first H16 with pn−3 = n/2 Day & Peterson’s conjecture is false K. & Mitrouli, 2007 g(16, H16 ) = 16
Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices
I
Gould, 1991 found a 13 × 13 matrix with growth 13.0205 The first part of Cryer’s conjecture is false. The second part still remains open
C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
I
Edelman & Mascarenhas, 1995 g(12, H12 ) = 12
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
I
I
Edelman & Friedman, 1998 found the first H16 with pn−3 = n/2 Day & Peterson’s conjecture is false K. & Mitrouli, 2007 g(16, H16 ) = 16
Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices
I
Gould, 1991 found a 13 × 13 matrix with growth 13.0205 The first part of Cryer’s conjecture is false. The second part still remains open
C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
I
Edelman & Mascarenhas, 1995 g(12, H12 ) = 12
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
I
I
Edelman & Friedman, 1998 found the first H16 with pn−3 = n/2 Day & Peterson’s conjecture is false K. & Mitrouli, 2007 g(16, H16 ) = 16
Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices
I
Gould, 1991 found a 13 × 13 matrix with growth 13.0205 The first part of Cryer’s conjecture is false. The second part still remains open
C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
I
Edelman & Mascarenhas, 1995 g(12, H12 ) = 12
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
I
I
Edelman & Friedman, 1998 found the first H16 with pn−3 = n/2 Day & Peterson’s conjecture is false K. & Mitrouli, 2007 g(16, H16 ) = 16
Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices
I
Gould, 1991 found a 13 × 13 matrix with growth 13.0205 The first part of Cryer’s conjecture is false. The second part still remains open
C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
I
Edelman & Mascarenhas, 1995 g(12, H12 ) = 12
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
I
I
Edelman & Friedman, 1998 found the first H16 with pn−3 = n/2 Day & Peterson’s conjecture is false K. & Mitrouli, 2007 g(16, H16 ) = 16
Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices
I
Gould, 1991 found a 13 × 13 matrix with growth 13.0205 The first part of Cryer’s conjecture is false. The second part still remains open
C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
I
Edelman & Mascarenhas, 1995 g(12, H12 ) = 12
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
I
I
Edelman & Friedman, 1998 found the first H16 with pn−3 = n/2 Day & Peterson’s conjecture is false K. & Mitrouli, 2007 g(16, H16 ) = 16
Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
Difficulty of the problem
On the growth factor for Hadamard matrices C. Kravvaritis Introduction
Pivot pattern invariant under H-equivalence operations, i.e. H-equivalent matrices may have different pivot patterns.
Gaussian Elimination Definitions Importance
History Determinants Preliminaries
A naive computer exhaustive search finding all possible H-equivalent H16 requires (16!)2 (216 )2 ≈ 1036 trials.
Solution The proposed idea Pivots from the beginning Pivots from the end
In addition, the pivot pattern of each one of these matrices should be computed. → many years of computations!
Numerical experiments Pivot patterns
SummaryReferences
Difficulty of the problem
On the growth factor for Hadamard matrices C. Kravvaritis Introduction
Pivot pattern invariant under H-equivalence operations, i.e. H-equivalent matrices may have different pivot patterns.
Gaussian Elimination Definitions Importance
History Determinants Preliminaries
A naive computer exhaustive search finding all possible H-equivalent H16 requires (16!)2 (216 )2 ≈ 1036 trials.
Solution The proposed idea Pivots from the beginning Pivots from the end
In addition, the pivot pattern of each one of these matrices should be computed. → many years of computations!
Numerical experiments Pivot patterns
SummaryReferences
Difficulty of the problem
On the growth factor for Hadamard matrices C. Kravvaritis Introduction
Pivot pattern invariant under H-equivalence operations, i.e. H-equivalent matrices may have different pivot patterns.
Gaussian Elimination Definitions Importance
History Determinants Preliminaries
A naive computer exhaustive search finding all possible H-equivalent H16 requires (16!)2 (216 )2 ≈ 1036 trials.
Solution The proposed idea Pivots from the beginning Pivots from the end
In addition, the pivot pattern of each one of these matrices should be computed. → many years of computations!
Numerical experiments Pivot patterns
SummaryReferences
Difficulty of the problem
On the growth factor for Hadamard matrices C. Kravvaritis Introduction
Pivot pattern invariant under H-equivalence operations, i.e. H-equivalent matrices may have different pivot patterns.
Gaussian Elimination Definitions Importance
History Determinants Preliminaries
A naive computer exhaustive search finding all possible H-equivalent H16 requires (16!)2 (216 )2 ≈ 1036 trials.
Solution The proposed idea Pivots from the beginning Pivots from the end
In addition, the pivot pattern of each one of these matrices should be computed. → many years of computations!
Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices
Solution Main idea 1: Calculate pivots with:
C. Kravvaritis
Lemma
Introduction
Let A be a CP matrix and A(j) denote the absolute value of the upper left j × j principal minor of A. (i) [Gantmacher 1959] The magnitude of the pivots appearing after application of GE operations on A is given by
Gaussian Elimination Definitions Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
pj =
A(j) , A(j − 1)
Pivots from the end
j = 1, 2, . . . , n,
A(0) = 1.
(1)
Numerical experiments Pivot patterns
SummaryReferences
(ii) [Cryer 1968] The maximum j × j leading principal minor of A, when the first j − 1 rows and columns are fixed, is A(j). → it is important to calculate minors!
On the growth factor for Hadamard matrices
Solution Main idea 1: Calculate pivots with:
C. Kravvaritis
Lemma
Introduction
Let A be a CP matrix and A(j) denote the absolute value of the upper left j × j principal minor of A. (i) [Gantmacher 1959] The magnitude of the pivots appearing after application of GE operations on A is given by
Gaussian Elimination Definitions Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
pj =
A(j) , A(j − 1)
Pivots from the end
j = 1, 2, . . . , n,
A(0) = 1.
(1)
Numerical experiments Pivot patterns
SummaryReferences
(ii) [Cryer 1968] The maximum j × j leading principal minor of A, when the first j − 1 rows and columns are fixed, is A(j). → it is important to calculate minors!
On the growth factor for Hadamard matrices
Solution Main idea 1: Calculate pivots with:
C. Kravvaritis
Lemma
Introduction
Let A be a CP matrix and A(j) denote the absolute value of the upper left j × j principal minor of A. (i) [Gantmacher 1959] The magnitude of the pivots appearing after application of GE operations on A is given by
Gaussian Elimination Definitions Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
pj =
A(j) , A(j − 1)
Pivots from the end
j = 1, 2, . . . , n,
A(0) = 1.
(1)
Numerical experiments Pivot patterns
SummaryReferences
(ii) [Cryer 1968] The maximum j × j leading principal minor of A, when the first j − 1 rows and columns are fixed, is A(j). → it is important to calculate minors!
On the growth factor for Hadamard matrices
Solution Main idea 1: Calculate pivots with:
C. Kravvaritis
Lemma
Introduction
Let A be a CP matrix and A(j) denote the absolute value of the upper left j × j principal minor of A. (i) [Gantmacher 1959] The magnitude of the pivots appearing after application of GE operations on A is given by
Gaussian Elimination Definitions Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
pj =
A(j) , A(j − 1)
Pivots from the end
j = 1, 2, . . . , n,
A(0) = 1.
(1)
Numerical experiments Pivot patterns
SummaryReferences
(ii) [Cryer 1968] The maximum j × j leading principal minor of A, when the first j − 1 rows and columns are fixed, is A(j). → it is important to calculate minors!
On the growth factor for Hadamard matrices C. Kravvaritis
First known effort for calculating the n − 1, n − 2 and n − 3 minors of Hadamard matrices:
Introduction Gaussian Elimination Definitions Importance
F. R. Sharpe, The maximum value of a determinant, Bull. Amer. Math. Soc. 14, 121–123 (1907)
History Determinants Preliminaries
n − 4 minors of Hadamard matrices, relative computer algorithm:
Solution The proposed idea Pivots from the beginning Pivots from the end
C. Koukouvinos, M. Mitrouli and J. Seberry, An algorithm to find formulae and values of minors of Hadamard matrices, Linear Algebra Appl. 330, 129–147 (2001)
Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices C. Kravvaritis
First known effort for calculating the n − 1, n − 2 and n − 3 minors of Hadamard matrices:
Introduction Gaussian Elimination Definitions Importance
F. R. Sharpe, The maximum value of a determinant, Bull. Amer. Math. Soc. 14, 121–123 (1907)
History Determinants Preliminaries
n − 4 minors of Hadamard matrices, relative computer algorithm:
Solution The proposed idea Pivots from the beginning Pivots from the end
C. Koukouvinos, M. Mitrouli and J. Seberry, An algorithm to find formulae and values of minors of Hadamard matrices, Linear Algebra Appl. 330, 129–147 (2001)
Numerical experiments Pivot patterns
SummaryReferences
Outline Introduction Gaussian Elimination Definitions Importance of this study
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
History of the problem Determinants Preliminary Results
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices
Preliminary Results
C. Kravvaritis Introduction
Lemma
Let A = (k − λ)Iv + λJv =
Gaussian Elimination
k λ ··· λ k ··· .. .. . . λ λ ···
λ λ , where k , λ k
are integers. Then,
Definitions Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end
v −1
det A = [k + (v − 1)λ](k − λ)
and for k 6= λ, −(v − 1)λ, A is nonsingular with k2
(2) A−1
=
1 {[k + (v − 1)λ]Iv − λJv }. (3) + (v − 2)k λ − (v − 1)λ2
Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions
Lemma Let B =
Importance
B1 B3
B2 B4
History
, B1 nonsingular. Then
det B = det B1 · det(B4 − B3 B1−1 B2 ).
Determinants Preliminaries
Solution
(4)
The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
Outline Introduction Gaussian Elimination Definitions Importance of this study
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
History of the problem Determinants Preliminary Results
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices C. Kravvaritis
Main idea 2: Calculation of pivots from the beginning and from the end with different techniques
Introduction Gaussian Elimination Definitions Importance
p1
. . p2 . . . p8 .. p9 .. p10 . . . p16
Determinants
−→
Solution
←−
History
Preliminaries
The proposed idea
and
Pivots from the beginning Pivots from the end Numerical experiments
det H p9 = Q16
i=1,i6=9 pi
Pivot patterns
SummaryReferences
Outline Introduction Gaussian Elimination Definitions Importance of this study
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
History of the problem Determinants Preliminary Results
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
Pivots from the beginning
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
I
we specify all possible j × j matrices that can appear as upper left corner of a CP H16 → algorithm Exist (symbolical, implemented in Maple)
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices
Preliminaries
C. Kravvaritis
u1 1 1 Uj = . .. 1 1
u2 . . . u2j−1 −1 u2j−1 1 ... 1 1 1 ... −1 −1 .. .. .. = [u 1 u 2 . . . u 2j−1 ] . . . 1 ... −1 −1 −1 . . . 1 −1
Introduction Gaussian Elimination Definitions Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
Lemma For the first j rows, j ≥ 3, of a normalized Hadamard matrix H of order n, n > 3, and for all the 2j−1 possible columns u 1 , . . . , u 2j−1 of Uj , it holds 0 ≤ ui ≤
n , for i = 1, . . . , 2j−1 . 4
SummaryReferences
On the growth factor for Hadamard matrices
Preliminaries
C. Kravvaritis
u1 1 1 Uj = . .. 1 1
u2 . . . u2j−1 −1 u2j−1 1 ... 1 1 1 ... −1 −1 .. .. .. = [u 1 u 2 . . . u 2j−1 ] . . . 1 ... −1 −1 −1 . . . 1 −1
Introduction Gaussian Elimination Definitions Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
Lemma For the first j rows, j ≥ 3, of a normalized Hadamard matrix H of order n, n > 3, and for all the 2j−1 possible columns u 1 , . . . , u 2j−1 of Uj , it holds 0 ≤ ui ≤
n , for i = 1, . . . , 2j−1 . 4
SummaryReferences
Implementation of algorithm Exist Step 1 We want to establish whether 1 1 1 1 1 −1 1 −1 H4 = 1 1 −1 −1 1 −1 −1 1
On the growth factor for Hadamard matrices C. Kravvaritis Introduction
Gaussian Elimination Definitions Importance
= [u u u u ] 1 6 4 7
History Determinants Preliminaries
Solution
always exists in the upper left 4 × 4 corner of a H16 .
The proposed idea Pivots from the beginning Pivots from the end
Step 2 u1 u2 u3 u4 u5 u6 u7 u8 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 U4 = 1 1 1 −1 −1 1 1 −1 −1 1 −1 1 −1 1 −1 1 −1
Numerical experiments Pivot patterns
SummaryReferences
Implementation of algorithm Exist Step 1 We want to establish whether 1 1 1 1 1 −1 1 −1 H4 = 1 1 −1 −1 1 −1 −1 1
On the growth factor for Hadamard matrices C. Kravvaritis Introduction
Gaussian Elimination Definitions Importance
= [u u u u ] 1 6 4 7
History Determinants Preliminaries
Solution
always exists in the upper left 4 × 4 corner of a H16 .
The proposed idea Pivots from the beginning Pivots from the end
Step 2 u1 u2 u3 u4 u5 u6 u7 u8 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 U4 = 1 1 1 −1 −1 1 1 −1 −1 1 −1 1 −1 1 −1 1 −1
Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices C. Kravvaritis
Step 3 counting of columns + orthogonality of every two distinct rows of U4 ⇒
Introduction Gaussian Elimination Definitions Importance
History
u1 + u2 + u3 + u4 + u5 + u6 + u7 + u8 u1 + u2 + u3 + u4 − u5 − u6 − u7 − u8 u1 + u2 − u3 − u4 + u5 + u6 − u7 − u8 u1 + u2 − u3 − u4 − u5 − u6 + u7 + u8 u1 − u2 + u3 − u4 + u5 − u6 + u7 − u8 u1 − u2 + u3 − u4 − u5 + u6 − u7 + u8 u1 − u2 − u3 + u4 + u5 − u6 − u7 + u8
=n =0 =0 =0 =0 =0 =0
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
solution: u1 u2 u3 u4 u5 u6 u7 u8
= 4 − u8 = u8 = u8 = 4 − u8 = u8 = 4 − u8 = 4 − u8 = u8
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
Lemma ⇒ u8 = 0, 1, 2, 3, 4 u8 = 0: (u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8 ) = (4, 0, 0, 4, 0, 4, 4, 0) u8 = 1: (u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8 ) = (3, 1, 1, 3, 1, 3, 3, 1) u8 = 2: (u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8 ) = (2, 2, 2, 2, 2, 2, 2, 2) u8 = 3: (u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8 ) = (1, 3, 3, 1, 3, 1, 1, 3) u8 = 4: (u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8 ) = (0, 4, 4, 0, 4, 0, 0, 4) → always u1 , u4 , u6 , u7 ≥ 1
Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
solution: u1 u2 u3 u4 u5 u6 u7 u8
= 4 − u8 = u8 = u8 = 4 − u8 = u8 = 4 − u8 = 4 − u8 = u8
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
Lemma ⇒ u8 = 0, 1, 2, 3, 4 u8 = 0: (u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8 ) = (4, 0, 0, 4, 0, 4, 4, 0) u8 = 1: (u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8 ) = (3, 1, 1, 3, 1, 3, 3, 1) u8 = 2: (u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8 ) = (2, 2, 2, 2, 2, 2, 2, 2) u8 = 3: (u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8 ) = (1, 3, 3, 1, 3, 1, 1, 3) u8 = 4: (u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8 ) = (0, 4, 4, 0, 4, 0, 0, 4) → always u1 , u4 , u6 , u7 ≥ 1
Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices
Results
C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
Table: j × j minors of H16
History Determinants
j 4 5 6 7 8
H16 (j) 16 32, 48 128, 160 256, 384, 512, 576 1024, 1536, 2048, 2304, 2560, 3072, 4096
Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
Outline Introduction Gaussian Elimination Definitions Importance of this study
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
History of the problem Determinants Preliminary Results
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices
Pivots from the end
C. Kravvaritis
I
Computation of (n − j) × (n − j) minors → algorithm Minors (symbolical, implemented in Maple)
Introduction Gaussian Elimination Definitions Importance
History
I
H=
M UjT
Uj D
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
I
solve the familiar linear system
Pivots from the end Numerical experiments Pivot patterns
I
form DD T partitioned appropriately in blocks
I
compute det DD T with consecutive applications of (4), with help of (2) and (3).
SummaryReferences
On the growth factor for Hadamard matrices
Pivots from the end
C. Kravvaritis
I
Computation of (n − j) × (n − j) minors → algorithm Minors (symbolical, implemented in Maple)
Introduction Gaussian Elimination Definitions Importance
History
I
H=
M UjT
Uj D
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
I
solve the familiar linear system
Pivots from the end Numerical experiments Pivot patterns
I
form DD T partitioned appropriately in blocks
I
compute det DD T with consecutive applications of (4), with help of (2) and (3).
SummaryReferences
On the growth factor for Hadamard matrices
Pivots from the end
C. Kravvaritis
I
Computation of (n − j) × (n − j) minors → algorithm Minors (symbolical, implemented in Maple)
Introduction Gaussian Elimination Definitions Importance
History
I
H=
M UjT
Uj D
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
I
solve the familiar linear system
Pivots from the end Numerical experiments Pivot patterns
I
form DD T partitioned appropriately in blocks
I
compute det DD T with consecutive applications of (4), with help of (2) and (3).
SummaryReferences
On the growth factor for Hadamard matrices
Pivots from the end
C. Kravvaritis
I
Computation of (n − j) × (n − j) minors → algorithm Minors (symbolical, implemented in Maple)
Introduction Gaussian Elimination Definitions Importance
History
I
H=
M UjT
Uj D
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
I
solve the familiar linear system
Pivots from the end Numerical experiments Pivot patterns
I
form DD T partitioned appropriately in blocks
I
compute det DD T with consecutive applications of (4), with help of (2) and (3).
SummaryReferences
On the growth factor for Hadamard matrices
Pivots from the end
C. Kravvaritis
I
Computation of (n − j) × (n − j) minors → algorithm Minors (symbolical, implemented in Maple)
Introduction Gaussian Elimination Definitions Importance
History
I
H=
M UjT
Uj D
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
I
solve the familiar linear system
Pivots from the end Numerical experiments Pivot patterns
I
form DD T partitioned appropriately in blocks
I
compute det DD T with consecutive applications of (4), with help of (2) and (3).
SummaryReferences
Outline Introduction Gaussian Elimination Definitions Importance of this study
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
History of the problem Determinants Preliminary Results
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
Numerical experiments
On the growth factor for Hadamard matrices C. Kravvaritis Introduction
Table: Values of minors of orders n − 1, . . . , n − 7 for Hadamard matrices of order n = 12, 16, 20
Gaussian Elimination Definitions Importance
History
order n−1 n−2 n−3 n−4 n−5 n−6 n−7
values of minors n/2−1
n 0, 2nn/2−2 0, 4nn/2−3 0, 8nn/2−4 , 16nn/2−4 0, 16nn/2−5 , 32nn/2−5 , 48nn/2−5 0, 32nn/2−6 , 64nn/2−6 , 96nn/2−6 , 128nn/2−6 , 160nn/2−6 0, 64nn/2−7 , 128nn/2−7 , 192nn/2−7 , 256nn/2−7 , 320nn/2−7 , 384nn/2−7 , 448nn/2−7 , 512nn/2−7 , 576nn/2−7
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices
The conjecture for minors of Hadamard matrices
C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
All possible (n − j) × (n − j), j ≥ 1, minors of Hadamard matrices are
History Determinants Preliminaries
0 or p · n
(n/2)−j
, for p = 2
j−1
j−1
,2 · 2
j−1
,3 · 2
j−1
,...,s · 2
,
Solution The proposed idea Pivots from the beginning Pivots from the end
where s · 2j−1 = max{det(A)|A ∈ Rj×j , with entries ± 1} and the value 0 is excluded from the case j = 1.
Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination
Table: Possible determinant values for n × n ±1 matrices
Definitions Importance
n 1 2 3 4 5 6 7
det 1 0, 2 0, 4 0, 8, 16 0, 16, 32, 48 0, 32, 64, 96, 128, 160 0, 64, 128, 192, 256, 320, 384, 448, 512, 576
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
Outline Introduction Gaussian Elimination Definitions Importance of this study
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History
History of the problem Determinants Preliminary Results
Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
Pivot patterns
On the growth factor for Hadamard matrices C. Kravvaritis
class I
II
III
IV/V
pivot pattern (1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16) (1,2,2,4,3,8/3,4,6,8/3,4,4,8,4,8,8,16) (1,2,2,4,3,8/3,2,4,4,4,4,8,8,8,8,16) (1,2,2,4,3,10/3,8/(10/3),4,16/3,5,16/(10/3),16/3,4,8,8,16) (1,2,2,4,2,4,4,4,4,4,4,8,4,8,8,16) (1,2,2,4,2,4,4,6,8/3,4,6,16/3,4,8,8,16) (1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16) (1,2,2,4,2,4,4,6,8/3,4,4,8,4,8,8,16) (1,2,2,4,2,4,4,4,9/2,16/(18/5),16/(10/3),16/3,4,8,8,16) (1,2,2,4,3,10/3,18/5,4,4,16/(18/5),16/(10/3),16/3,4,8,8,16) (1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16) (1,2,2,4,2,4,4,4,9/2,16/(18/5),16/(10/3),16/3,4,8,8,16)
Introduction Gaussian Elimination Definitions Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices C. Kravvaritis
Appropriate substitution of minors in (1) → we obtain all 34 possible pivot patterns of H16
Introduction Gaussian Elimination Definitions Importance
History
For a CP matrix A:
Determinants
g(n, A) =
(r −1) max1≤r ≤n |arr |
|a11 |
Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
Hence g(16, H16 ) = 16
SummaryReferences
Conclusions-Discussions-Open Problems
On the growth factor for Hadamard matrices C. Kravvaritis
I
pivot patterns of H20 , H24 etc.
Introduction Gaussian Elimination
I
High complexity → more effective implementation
Definitions Importance
History
I
Parallel implementation of the two main independent tasks
Determinants Preliminaries
Solution The proposed idea
I
Classification of the pivot patterns of H16
I
the value 8 as fourth pivot from the end for H16
Pivots from the beginning Pivots from the end Numerical experiments
I
sharper bound than n/4 (maybe n/8)
I
connection of Hadamard minors with ±1 determinants
Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices
I
I
Statistical approach L. N. Trefethen and R. S. Schreiber, Average-case stability of Gaussian elimination, SIAM J. Matrix Anal. Appl. 11, 335–360 (1990) Generalization for OD’s: An orthogonal design (OD) of order n and type (u1 , u2 , . . . , ut ), ui positive integers, is an n × n matrix D with entries from the set {0, ±x1 , ±x2 , . . . , ±xt } that satisfies ! t X T T 2 DD = D D = ui xi In . i=1
C. Kravvaritis Introduction Gaussian Elimination Definitions Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
References Progress in the growth factor for Hadamard matrices: L. Tornheim, Pivot size in Gauss reduction, Tech. Report, Calif. Res. Corp., Richmond, Calif., February 1964.
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination
C. W. Cryer, Pivot size in Gaussian elimination, Numer. Math. 12, 335–345 (1968) J. Day and B. Peterson, Growth in Gaussian Elimination, Amer. Math. Monthly 95, 489–513 (1988)
Definitions Importance
History Determinants Preliminaries
Solution The proposed idea
A. Edelman and W. Mascarenhas, On the complete pivoting conjecture for a Hadamard matrix of order 12, Linear Multilinear Algebra 38, 181–187 (1995) A. Edelman and D. Friedman, A counterexample to a Hadamard matrix pivot conjecture, Linear Multilinear Algebra 44, 53–56 (1998) C. Kravvaritis and M. Mitrouli, Evaluation of Minors associated to weighing matrices, Linear Algebra Appl. 426, 774-809 (2007)
Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
References Progress in the growth factor generally: A.M. Cohen, A note on pivot size in Gaussian elimination, Linear Algebra Appl. 8, 361–368 (1974)
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination
T. A. Driscoll and K. L. Maki, Searching for Rare Growth Factors Using Multicanonical Monte Carlo Methods, SIAM Review 49, 673–692 (2007)
Definitions Importance
History Determinants Preliminaries
A. Edelman, The Complete Pivoting Conjecture for Gaussian Elimination is false, The Mathematica Journal 2, 58–61 (1992)
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
N. Gould, On growth in Gaussian elimination with pivoting, SIAM J. Matrix Anal. Appl. 12, 354–361 (1991) N. J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, 2002. N. J. Higham and D. J. Higham, Large growth factors in Gaussian Elimination with Pivoting, SIAM J. Matrix Anal. Appl. 10 155–164 (1989)
SummaryReferences
References
On the growth factor for Hadamard matrices C. Kravvaritis
Books on orthogonal matrices:
Introduction Gaussian Elimination Definitions
A. V. Geramita and J. Seberry, Orthogonal Designs: Quadratic Forms and Hadamard Matrices, Marcel Dekker, New York-Basel (1979) A. S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays: Theory and Application, Springer, New York, 1999. K. J. Horadam, Hadamard matrices and their appplications, Princeton University Press, Princeton (2007)
Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
On the growth factor for Hadamard matrices C. Kravvaritis Introduction Gaussian Elimination Definitions
Thank you very much for your attention!
Importance
History Determinants Preliminaries
Solution The proposed idea Pivots from the beginning
http://ckravv.googlepages.com
Pivots from the end Numerical experiments Pivot patterns
SummaryReferences
