Efficient Calculation of Determinants of Symbolic
Matrices with Many Variables
Ziv Scully
Second Annual MIT PRIMES Conference
May 20, 2012
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
1 / 12
Background
Vectors and Volumes
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
2 / 12
Background
Vectors and Volumes
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
2 / 12
Background
Vectors and Volumes
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
2 / 12
Background
Vectors and Volumes
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
2 / 12
Background
Vectors and Volumes
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
2 / 12
Background
Determinants
2 × 2 matrices:
Ziv Scully (MIT PRIMES)
�
a b
det
c d
�
= ad − bc
Determinants with Many Variables
May 20, 2012
3 / 12
Background
Determinants
2 × 2 matrices:
Ziv Scully (MIT PRIMES)
�
a b
det
c d
�
= ad − bc
Determinants with Many Variables
May 20, 2012
3 / 12
Background
Determinants
2 × 2 matrices:
Ziv Scully (MIT PRIMES)
�
a b
det
c d
�
= ad − bc
Determinants with Many Variables
May 20, 2012
3 / 12
Background
Determinants
2 × 2 matrices:
�
a b
det
c d
�
= ad − bc
3 × 3 matrices:


a b c
det d e f  = aei + bf g + cdh − af h − bdi − ceg
g h i
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
3 / 12
Background
Determinants
2 × 2 matrices:
�
a b
det
c d
�
= ad − bc
3 × 3 matrices:


a b c
det d e f  = aei + bf g + cdh − af h − bdi − ceg
g h i
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
3 / 12
Background
Determinants
2 × 2 matrices:
�
a b
det
c d
�
= ad − bc
3 × 3 matrices:


a b c
det d e f  = aei + bf g + cdh − af h − bdi − ceg
g h i
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
3 / 12
Background
Determinants
2 × 2 matrices:
�
a b
det
c d
�
= ad − bc
3 × 3 matrices:


a b c
det d e f  = aei + bf g + cdh − af h − bdi − ceg
g h i
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
3 / 12
Background
Determinants
2 × 2 matrices:
�
a b
det
c d
�
= ad − bc
3 × 3 matrices:


a b c
det d e f  = aei + bf g + cdh − af h − bdi − ceg
g h i
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
3 / 12
Background
Determinants
2 × 2 matrices:
�
a b
det
c d
�
= ad − bc
3 × 3 matrices:


a b c
det d e f  = aei + bf g + cdh − af h − bdi − ceg
g h i
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
3 / 12
Background
Determinants
2 × 2 matrices:
�
a b
det
c d
�
= ad − bc
3 × 3 matrices:


a b c
det d e f  = aei + bf g + cdh − af h − bdi − ceg
g h i
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
3 / 12
Background
Determinants
2 × 2 matrices:
�
a b
det
c d
�
= ad − bc
3 × 3 matrices:


a b c
det d e f  = aei + bf g + cdh − af h − bdi − ceg
g h i
n × n matrices:
det (A) =
��
σ∈Sn
Ziv Scully (MIT PRIMES)
sgn(σ)
n
�
Ai,σ(i)
i=1
Determinants with Many Variables
�
May 20, 2012
3 / 12
Background
Motivation
Linear systems.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
4 / 12
Background
Motivation
Linear systems.
Calculus.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
4 / 12
Background
Motivation
Linear systems.
Calculus.
Control theory.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
4 / 12
Background
Motivation
Linear systems.
Calculus.
Control theory.
Engineering models.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
4 / 12
Background
Motivation
Linear systems.
Calculus.
Control theory.
Engineering models.
Code generation.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
4 / 12
Background
Motivation
Linear systems.
Calculus.
Control theory.
Engineering models.
Code generation.
We are interested in matrices with polynomial entries.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
4 / 12
Algorithms
Minor Expansion
Naive calculation requires Θ(n! n) polynomial multiplications.

a
e
det 
i
m
b
f
j
n

af kp − af lo − agjp + agln + ahjo − ahkn
c d

− bekp + belo + bgip − bglm − bhio + bhkm
g h
=

k l
+ cejp − celn − cf ip + cf lm + chin − chjm
o p
− dejo + dekn + df io − df km − dgin + dgjm
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
5 / 12
Algorithms
Minor Expansion
Naive calculation requires Θ(n! n) polynomial multiplications.

a
e
det 
i
m
b
f
j
n

a(f kp − f lo − gjp + gln + hjo − hkn)
c d

− b(ekp − elo − gip + glm + hio − hkm)
g h
=

k l
+ c(ejp − eln − f ip + f lm + hin − hjm)
o p
− d(ejo − ekn − f io + f km + gin − gjm)
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
5 / 12
Algorithms
Minor Expansion
Naive calculation requires Θ(n! n) polynomial multiplications.

a
e
det 
i
m
b
f
j
n

a(f kp − f lo − gjp + gln + hjo − hkn)
c d

− b(ekp − elo − gip + glm + hio − hkm)
g h
=

k l
+ c(ejp − eln − f ip + f lm + hin − hjm)
o p
− d(ejo − ekn − f io + f km + gin − gjm)
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
5 / 12
Algorithms
Minor Expansion
Naive calculation requires Θ(n! n) polynomial multiplications.

a
e
det 
i
m
b
f
j
n

a(f kp − f lo − gjp + gln + hjo − hkn)
c d

− b(ekp − elo − gip + glm + hio − hkm)
g h
=

k l
+ c(ejp − eln − f ip + f lm + hin − hjm)
o p
− d(ejo − ekn − f io + f km + gin − gjm)
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
5 / 12
Algorithms
Minor Expansion
Naive calculation requires Θ(n! n) polynomial multiplications.

a
e
det 
i
m
b
f
j
n

a(f kp − f lo − gjp + gln + hjo − hkn)
c d

− b(ekp − elo − gip + glm + hio − hkm)
g h
=

k l
+ c(ejp − eln − f ip + f lm + hin − hjm)
o p
− d(ejo − ekn − f io + f km + gin − gjm)
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
5 / 12
Algorithms
Minor Expansion
Naive calculation requires Θ(n! n) polynomial multiplications.

a
e
det 
i
m
b
f
j
n

a(f kp − f lo − gjp + gln + hjo − hkn)
c d

− b(ekp − elo − gip + glm + hio − hkm)
g h
=

k l
+ c(ejp − eln − f ip + f lm + hin − hjm)
o p
− d(ejo − ekn − f io + f km + gin − gjm)
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
5 / 12
Algorithms
Minor Expansion
Naive calculation requires Θ(n! n) polynomial multiplications.

a
e
det 
i
m
b
f
j
n

a(f kp − f lo − gjp + gln + hjo − hkn)
c d

− b(ekp − elo − gip + glm + hio − hkm)
g h
=

k l
+ c(ejp − eln − f ip + f lm + hin − hjm)
o p
− d(ejo − ekn − f io + f km + gin − gjm)
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
5 / 12
Algorithms
Minor Expansion
Naive calculation requires Θ(n! n) polynomial multiplications.

a
e
det 
i
m
b
f
j
n

a(f (kp − lo) − g(jp − ln) + h(jo − kn))
c d

− b(e(kp − lo) − g(ip − lm) + h(io − km))
g h
=

k l
+ c(e(jp − ln) − f (ip − lm) + h(in − jm))
o p
− d(e(jo − kn) − f (io − km) + g(in − jm))
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
5 / 12
Algorithms
Minor Expansion
Naive calculation requires Θ(n! n) polynomial multiplications.

a
e
det 
i
m
b
f
j
n

a(f (kp − lo) − g(jp − ln) + h(jo − kn))
c d

− b(e(kp − lo) − g(ip − lm) + h(io − km))
g h
=

k l
+ c(e(jp − ln) − f (ip − lm) + h(in − jm))
o p
− d(e(jo − kn) − f (io − km) + g(in − jm))
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
5 / 12
Algorithms
Minor Expansion
Naive calculation requires Θ(n! n) polynomial multiplications.

a
e
det 
i
m
b
f
j
n

a(f (kp − lo) − g(jp − ln) + h(jo − kn))
c d

− b(e(kp − lo) − g(ip − lm) + h(io − km))
g h
=

k l
+ c(e(jp − ln) − f (ip − lm) + h(in − jm))
o p
− d(e(jo − kn) − f (io − km) + g(in − jm))
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
5 / 12
Algorithms
Minor Expansion
Naive calculation requires Θ(n! n) polynomial multiplications.

a
e
det 
i
m
b
f
j
n

a(f (kp − lo) − g(jp − ln) + h(jo − kn))
c d

− b(e(kp − lo) − g(ip − lm) + h(io − km))
g h
=

k l
+ c(e(jp − ln) − f (ip − lm) + h(in − jm))
o p
− d(e(jo − kn) − f (io − km) + g(in − jm))
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
5 / 12
Algorithms
Minor Expansion
Naive calculation requires Θ(n! n) polynomial multiplications.

a
e
det 
i
m
b
f
j
n

a(f (kp − lo) − g(jp − ln) + h(jo − kn))
c d

− b(e(kp − lo) − g(ip − lm) + h(io − km))
g h
=

k l
+ c(e(jp − ln) − f (ip − lm) + h(in − jm))
o p
− d(e(jo − kn) − f (io − km) + g(in − jm))
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
5 / 12
Algorithms
Minor Expansion
Naive calculation requires Θ(n! n) polynomial multiplications.

a
e
det 
i
m
b
f
j
n

a(f (kp − lo) − g(jp − ln) + h(jo − kn))
c d

− b(e(kp − lo) − g(ip − lm) + h(io − km))
g h
=

k l
+ c(e(jp − ln) − f (ip − lm) + h(in − jm))
o p
− d(e(jo − kn) − f (io − km) + g(in − jm))
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
5 / 12
Algorithms
Minor Expansion
Naive calculation requires Θ(n! n) polynomial multiplications.

a
e
det 
i
m
b
f
j
n

a(f (kp − lo) − g(jp − ln) + h(jo − kn))
c d

− b(e(kp − lo) − g(ip − lm) + h(io − km))
g h
=

k l
+ c(e(jp − ln) − f (ip − lm) + h(in − jm))
o p
− d(e(jo − kn) − f (io − km) + g(in − jm))
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
5 / 12
Algorithms
Minor Expansion
Naive calculation requires Θ(n! n) polynomial multiplications.

a
e
det 
i
m
b
f
j
n

a(f (kp − lo) − g(jp − ln) + h(jo − kn))
c d

− b(e(kp − lo) − g(ip − lm) + h(io − km))
g h
=

k l
+ c(e(jp − ln) − f (ip − lm) + h(in − jm))
o p
− d(e(jo − kn) − f (io − km) + g(in − jm))
n � �
�
n
Minor expansion requires
i
∈ Θ(2n n) polynomial multiplications.
i
i=2
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
5 / 12
Algorithms
Gaussian Elimination

a
0
det 
0
0
Ziv Scully (MIT PRIMES)

b c d
f g h
 = af kp
0 k l
0 0 p
Determinants with Many Variables
May 20, 2012
6 / 12
Algorithms
Gaussian Elimination

a
0
det 
0
0
Ziv Scully (MIT PRIMES)

b c d
f g h
 = af kp
0 k l
0 0 p
Determinants with Many Variables
May 20, 2012
6 / 12
Algorithms
Gaussian Elimination


a b c ...


det  e f g . . . =
.. .. .. . .
.
. . .
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
6 / 12
Algorithms
Gaussian Elimination



a
b
c
...
a b c ...
e
e
e




det  e f g . . . = det e − a a f − a b g − a c . . .
..
..
..
.. .. .. . .
..
.
.
.
.
.
. . .

Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
6 / 12
Algorithms
Gaussian Elimination


a b c ...



det  e f g . . . = det 
.. .. .. . .
.
. . .

Ziv Scully (MIT PRIMES)
a
0
..
.

b
c
...
f − ae b g − ae c . . .

..
..
..
.
.
.
Determinants with Many Variables
May 20, 2012
6 / 12
Algorithms
Gaussian Elimination


a b c ...



det  e f g . . . = det 
.. .. .. . .
.
. . .

a
0
..
.

b
c
...
f − ae b g − ae c . . .

..
..
..
.
.
.
A(1) = A,
(k)
(k+1)
Ai,j
=
(k)
Ai,j
Ziv Scully (MIT PRIMES)
−
Ai,k
(k)
Ak,k
(k)
Ak,j
Determinants with Many Variables
May 20, 2012
6 / 12
Algorithms
Gaussian Elimination


a b c ...



det  e f g . . . = det 
.. .. .. . .
.
. . .
a
0
..
.
A(1) = A,
A(1) = A,


b
c
...
f − ae b g − ae c . . .

..
..
..
.
.
.
(k)
(k+1)
Ai,j
=
(k)
Ai,j
Ziv Scully (MIT PRIMES)
−
Ai,k
(k)
(k)
A
(k) k,j
Ak,k
(k+1)
Ai,j
=
Determinants with Many Variables
(0)
A0,0 = 1,
(k)
(k)
(k)
Ai,j Ak,k − Ai,k Ak,j
(k−1)
Ak−1,k−1
May 20, 2012
6 / 12
Algorithms
Gaussian Elimination


a b c ...



det  e f g . . . = det 
.. .. .. . .
.
. . .
a
0
..
.
A(1) = A,
A(1) = A,


b
c
...
f − ae b g − ae c . . .

..
..
..
.
.
.
(k)
(k+1)
Ai,j
=
(k)
Ai,j
−
Ai,k
(k)
(k)
A
(k) k,j
Ak,k
(k+1)
Ai,j
Fraction-free Gaussian elimination requires
multiplications and divisions.
Ziv Scully (MIT PRIMES)
=
n
�
i=1
(0)
A0,0 = 1,
(k)
(k)
(k)
Ai,j Ak,k − Ai,k Ak,j
(k−1)
Ak−1,k−1
Θ(i2 ) ∈ Θ(n3 ) polynomial
Determinants with Many Variables
May 20, 2012
6 / 12
Algorithms
Comparison
Preservation of “simple” polynomials (e.g., those with few terms):
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
7 / 12
Algorithms
Comparison
Preservation of “simple” polynomials (e.g., those with few terms):
Minor expansion: Entries are preserved.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
7 / 12
Algorithms
Comparison
Preservation of “simple” polynomials (e.g., those with few terms):
Minor expansion: Entries are preserved.
Gaussian elimination: Entries made more complicated each step.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
7 / 12
Algorithms
Comparison
Preservation of “simple” polynomials (e.g., those with few terms):
Minor expansion: Entries are preserved.
Gaussian elimination: Entries made more complicated each step.
�
Consider an n × n matrix with entries of the form si=1 ai xi :
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
7 / 12
Algorithms
Comparison
Preservation of “simple” polynomials (e.g., those with few terms):
Minor expansion: Entries are preserved.
Gaussian elimination: Entries made more complicated each step.
�
Consider an n × n matrix with entries of the form si=1 ai xi :
cost of ME
cost of FFGE
cost ratio =
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
7 / 12
Algorithms
Row Permutation
Let p, q and r be polynomials with a, b and c terms, respectively, with no
variables in common.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
8 / 12
Algorithms
Row Permutation
Let p, q and r be polynomials with a, b and c terms, respectively, with no
variables in common.
(pq)r requires ab + abc integer multiplications.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
8 / 12
Algorithms
Row Permutation
Let p, q and r be polynomials with a, b and c terms, respectively, with no
variables in common.
(pq)r requires ab + abc integer multiplications.
(pr)q requires ac + abc integer multiplications.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
8 / 12
Algorithms
Row Permutation
Let p, q and r be polynomials with a, b and c terms, respectively, with no
variables in common.
(pq)r requires ab + abc integer multiplications.
(pr)q requires ac + abc integer multiplications.
(qr)p requires bc + abc integer multiplications.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
8 / 12
Algorithms
Row Permutation
Let p, q and r be polynomials with a, b and c terms, respectively, with no
variables in common.
(pq)r requires ab + abc integer multiplications.
(pr)q requires ac + abc integer multiplications.
(qr)p requires bc + abc integer multiplications.
We want to defer multiplying by polynomials with many terms.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
8 / 12
Algorithms
Row Permutation
Let p, q and r be polynomials with a, b and c terms, respectively, with no
variables in common.
(pq)r requires ab + abc integer multiplications.
(pr)q requires ac + abc integer multiplications.
(qr)p requires bc + abc integer multiplications.
We want to defer multiplying by polynomials with many terms.
Absolute value of determinant is invariant under row swaps.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
8 / 12
Empirical Results
Experiment Setup
Random polynomial matrices:
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
9 / 12
Empirical Results
Experiment Setup
Random polynomial matrices:
9 × 9.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
9 / 12
Empirical Results
Experiment Setup
Random polynomial matrices:
9 × 9.
5 variables.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
9 / 12
Empirical Results
Experiment Setup
Random polynomial matrices:
9 × 9.
5 variables.
For various values of p: an entry is 0 with probability p and has
between one and four terms otherwise. (Any number is equally likely.)
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
9 / 12
Empirical Results
Experiment Setup
Random polynomial matrices:
9 × 9.
5 variables.
For various values of p: an entry is 0 with probability p and has
between one and four terms otherwise. (Any number is equally likely.)
Each term is constant or linear in each variable.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
9 / 12
Empirical Results
Experiment Setup
Random polynomial matrices:
9 × 9.
5 variables.
For various values of p: an entry is 0 with probability p and has
between one and four terms otherwise. (Any number is equally likely.)
Each term is constant or linear in each variable.
Sort rows r in ascending order based on these scores:
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
9 / 12
Empirical Results
Experiment Setup
Random polynomial matrices:
9 × 9.
5 variables.
For various values of p: an entry is 0 with probability p and has
between one and four terms otherwise. (Any number is equally likely.)
Each term is constant or linear in each variable.
Sort rows r in ascending order based on these scores:
Control, no sorting.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
9 / 12
Empirical Results
Experiment Setup
Random polynomial matrices:
9 × 9.
5 variables.
For various values of p: an entry is 0 with probability p and has
between one and four terms otherwise. (Any number is equally likely.)
Each term is constant or linear in each variable.
Sort rows r in ascending order based on these scores:
Control, no sorting.
Number of nonzero entries in a row,
Ziv Scully (MIT PRIMES)
�n
i=1 (ri
Determinants with Many Variables
�= 0).
May 20, 2012
9 / 12
Empirical Results
Experiment Setup
Random polynomial matrices:
9 × 9.
5 variables.
For various values of p: an entry is 0 with probability p and has
between one and four terms otherwise. (Any number is equally likely.)
Each term is constant or linear in each variable.
Sort rows r in ascending order based on these scores:
Control, no sorting.
�
Number of nonzero entries in a row, ni=1 (ri �= 0).
�
Total number of terms in a row, ni=1 nterms(ri ).
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
9 / 12
Empirical Results
Experiment Setup
Random polynomial matrices:
9 × 9.
5 variables.
For various values of p: an entry is 0 with probability p and has
between one and four terms otherwise. (Any number is equally likely.)
Each term is constant or linear in each variable.
Sort rows r in ascending order based on these scores:
Control, no sorting.
�
Number of nonzero entries in a row, ni=1 (ri �= 0).
�
Total number of terms in a row, ni=1 nterms(ri ).
Product
of one more than number of terms for each entry of a row,
�n
(nterms(r
i ) + 1).
i=1
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
9 / 12
Empirical Results
Data
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
10 / 12
Empirical Results
Data
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
10 / 12
Conclusion
Further Questions
Define “complexity” of matrix entries:
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
11 / 12
Conclusion
Further Questions
Define “complexity” of matrix entries:
Number of terms.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
11 / 12
Conclusion
Further Questions
Define “complexity” of matrix entries:
Number of terms.
Variables present.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
11 / 12
Conclusion
Further Questions
Define “complexity” of matrix entries:
Number of terms.
Variables present.
Context of a row.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
11 / 12
Conclusion
Further Questions
Define “complexity” of matrix entries:
Number of terms.
Variables present.
Context of a row.
Investigate specific types of matrices:
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
11 / 12
Conclusion
Further Questions
Define “complexity” of matrix entries:
Number of terms.
Variables present.
Context of a row.
Investigate specific types of matrices:
Bézout matrices.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
11 / 12
Conclusion
Further Questions
Define “complexity” of matrix entries:
Number of terms.
Variables present.
Context of a row.
Investigate specific types of matrices:
Bézout matrices.
Do more experiments!
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
11 / 12
Conclusion
Further Questions
Define “complexity” of matrix entries:
Number of terms.
Variables present.
Context of a row.
Investigate specific types of matrices:
Bézout matrices.
Do more experiments!
Vary other criteria.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
11 / 12
Conclusion
Further Questions
Define “complexity” of matrix entries:
Number of terms.
Variables present.
Context of a row.
Investigate specific types of matrices:
Bézout matrices.
Do more experiments!
Vary other criteria.
Different algorithms and variations.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
11 / 12
Conclusion
Further Questions
Define “complexity” of matrix entries:
Number of terms.
Variables present.
Context of a row.
Investigate specific types of matrices:
Bézout matrices.
Do more experiments!
Vary other criteria.
Different algorithms and variations.
Crossover points between algorithms.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
11 / 12
Conclusion
Further Questions
Define “complexity” of matrix entries:
Number of terms.
Variables present.
Context of a row.
Investigate specific types of matrices:
Bézout matrices.
Do more experiments!
Vary other criteria.
Different algorithms and variations.
Crossover points between algorithms.
Machine learning.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
11 / 12
Conclusion
Acknowledgments
Dr. Tanya Khovanova, MIT, for mentoring me.
Dr. Stefan Wehmeier, MathWorks, for suggesting the project and
introducing me to the field.
Dr. Ben Hinkle, MathWorks, for arranging software license and an
international conference call.
MIT PRIMES, for giving me this unique research opportunity.
The MathWorks, Inc., for providing software and supporting the
research.
Ziv Scully (MIT PRIMES)
Determinants with Many Variables
May 20, 2012
12 / 12