Use of Computer Technology for
Insight and Proof
Strengths, Weaknesses and Practical
Strategies
(i) The role of CAS in analysis
(ii) Four practical mechanisms
(iii) Applications
Kent Pearce
Texas Tech University
Presentation: Fresno, California, 24 September 2010
Question
 Consider
f ( x)  cos x
g ( x)  2 e

x
Question
 Consider
f ( x)  cos x
g ( x)  2 e

x
Question
 Consider
h  2 e  cos( x)

x
Question
 Consider
h  2 e  cos( x)

x
Question
 Given a function f on an interval [a, b], what
does it take to show that f is non-negative on
[a, b]?
Transcendental Functions
 Consider
g ( x)  cos( x)
Transcendental Functions
 Consider
g ( x)  cos( x)
Transcendental Functions
cos(0)
1
cos(0.95)
0.5816830895
cos(0.95 + 2000000000*π)
0.5816830895
cos(0.95 + 2000000000.*π)
cos(0.95 + 2000000000.*π)
Blackbox Approximations
 Transcendental / Special Functions
Polynomials/Rational Functions
 CAS Calculations

Integer Arithmetic

Rational Values vs Irrational Values
 Floating Point Calculation
Question
 Given a function f on an interval [a, b], what
does it take to show that f is non-negative on
[a, b]?
(P)Lots of Dots
(P)Lots of Dots
(P)Lots of Dots
(P)Lots of Dots
y  f ( x) 
1
2x 1
(P)Lots of Dots
y  f ( x) 
1
2x 1
Question
 Given a function f on an interval [a, b], what
does it take to show that f is non-negative on
[a, b]?
 Proof by Picture

Maple, Mathematica, Matlab, Mathcad,
Excel, Graphing Calculators, Java Applets
Practical Methods
 A. Sturm Sequence Arguments
 B. Linearity / Monotonicity Arguments
 C. Special Function Estimates
 D. Grid Estimates
Applications
 "On a Coefficient Conjecture of Brannan," Complex
Variables. Theory and Application. An International
Journal 33 (1997) 51_61, with Roger W. Barnard and
William Wheeler.
 "A Sharp Bound on the Schwarzian Derivatives of
Hyperbolically Convex Functions," Proceeding of the
London Mathematical Society 93 (2006), 395_417, with
Roger W. Barnard, Leah Cole and G. Brock Williams.
 "The Verification of an Inequality," Proceedings of the
International Conference on Geometric Function Theory,
Special Functions and Applications (ICGFT) (accepted)
with Roger W. Barnard.
 "Iceberg-Type Problems in Two Dimensions," with
Roger.W. Barnard and Alex.Yu. Solynin
Practical Methods
 A. Sturm Sequence Arguments
 B. Linearity / Monotonicity Arguments
 C. Special Function Estimates
 D. Grid Estimates
Iceberg-Type Problems
Iceberg-Type Problems
 Dual Problem for Class 0
Let D= {z  : 0  | z |  1} and let
1

  { f ( z )   a0  a1 z 
: f is analytic,
z
univalent on D}. For f   let E f  \ f (D)
and 0  { f   | 0  E f }. For 0 < h < 4, let
H h  {z | Re( z )  h}. Find
A(h)  max area( E f  H h )
f 0
Iceberg-Type Problems
 Extremal Configuration




Symmetrization
Polarization
Variational Methods
Boundary Conditions
Iceberg-Type Problems
Iceberg-Type Problems
 We obtained explicit formulas for A = A(r)
and h = h(r). To show that we could write
A = A(h), we needed to show that h = h(r) was
monotone.
Practical Methods
 A. Sturm Sequence Arguments
 B. Linearity / Monotonicity Arguments
 C. Special Function Estimates
 D. Grid Estimates
Sturm Sequence Arguments
 General theorem for counting the number of
distinct roots of a polynomial f on an interval
(a, b)
 N. Jacobson, Basic Algebra. Vol. I., pp. 311315,W. H. Freeman and Co., New York, 1974.
 H. Weber, Lehrbuch der Algebra, Vol. I., pp. 301313, Friedrich Vieweg und Sohn, Braunschweig,
1898
Sturm Sequence Arguments
 Sturm’s Theorem. Let f be a non-constant
polynomial with rational coefficients and let a < b
be rational numbers. Let S f  { f 0 , f1 , , f s }
be the standard sequence for f . Suppose that
f (a)  0, f (b)  0. Then, the number of distinct
roots of f on (a, b) is Va  Vb where Vc denotes
the number of sign changes of
S f (c)  { f 0 (c), f1 (c),
, f s (c)}
Sturm Sequence Arguments
 Sturm’s Theorem (Generalization). Let f be a
non-constant polynomial with rational coefficients
and let a < b be rational numbers. Let
S f  { f 0 , f1 , , f s } be the standard sequence for
f . Suppose that f (a)  0, f (b)  0. Then, the
number of distinct roots of f on (a, b] is Va  Vb
where Vc denotes the number of sign changes of
S f (c)  { f 0 (c), f1 (c),
, f s (c)}
Sturm Sequence Arguments
 For a given f, the standard sequence S f is
constructed as:
f0  f
f1  f 
f 2 : f 0  f1q1  f 2
f 3 : f1  f 2 q2  f 3
Sturm Sequence Arguments
 Polynomial
Sturm Sequence Arguments
 Polynomial
Linearity / Monotonicity
 Consider
f ( x, Z )  c0 ( x)  c1 ( x)Z
where   Z  
Let
f ( x)  f ( x, Z ) Z   c0  c1 ,
f  ( x)  f ( x, Z ) Z    c0  c1
Then,
min { f ( x), f  ( x)}  f ( x, Z )  max{ f ( x), f  ( x)}
x( a ,b )
x( a ,b )
Iceberg-Type Problems
 We obtained explicit formulas for A = A(r)
and h = h(r). To show that we could write
A = A(h), we needed to show that h = h(r) was
monotone.
Iceberg-Type Problems
 From the construction we explicitly found
where
Iceberg-Type Problems
Iceberg-Type Problems
where
Iceberg-Type Problems
 It remained to show
g  g (r )  (c0  c1P)  (d0  d1P)Q
was non-negative. In a separate lemma, we
showed 0 < Q < 1. Hence, using the linearity of
Q in g, we needed to show
g 0  (c0  c1P)  (d 0  d1P)  0
g1  (c0  c1 P)  (d 0  d1P) 1
were non-negative
Iceberg-Type Problems
 In a second lemma, we showed s < P < t where
Let
g0,s  g0 Ps , g0,t  g0 Pt , g1, s  g1 Ps , g1,t  g1 Pt .
Each g0, s , g0,t , g1, s , g1,t is a polynomial with
rational coefficients for which a Sturm sequence
argument show that it is non-negative.
Practical Methods
 A. Sturm Sequence Arguments
 B. Linearity / Monotonicity Arguments
 C. Special Function Estimates
 D. Grid Estimates
Notation & Definitions

D  {z :| z |  1}
Notation & Definitions

D  {z :| z |  1}
hyperbolic metric
2 | dz |
 ( z ) | dz |
2
1 | z |
Notation & Definitions

D  {z :| z |  1}
 Hyberbolic Geodesics
Notation & Definitions

D  {z :| z |  1}
 Hyberbolic Geodesics
 Hyberbolically Convex Set
Notation & Definitions

D  {z :| z |  1}
 Hyberbolic Geodesics
 Hyberbolically Convex Set
 Hyberbolically Convex Function
Notation & Definitions

D  {z :| z |  1}
 Hyberbolic Geodesics
 Hyberbolically Convex Set
 Hyberbolically Convex Function
 Hyberbolic Polygon
o Proper Sides
Examples
k ( z ) 
2 z
(1  z )  (1  z ) 2  4 2 z
k
Examples

f ( z )  tan  

where  
z
 (1  2
2
1
4  2
cos 2   )
0

2 , 0  
2
K (cos )
f

d 

Schwarz Norm || S f ||D
For f  A(D) let

Sf  

2

f   1  f  
  

f   2 f  
and
|| S f ||D  sup{D2 ( z) | S f ( z) |: z D}
where
1
D ( z ) 
1 | z |2
Extremal Problems for || S f ||D
 Euclidean Convexity

Nehari (1976):
f (D) convex  || S f ||D  2
Extremal Problems for || S f ||D
 Euclidean Convexity

Nehari (1976):
f (D) convex  || S f ||D  2
 Spherical Convexity

Mejía, Pommerenke (2000):
f (D) convex  || S f ||D  2
Extremal Problems for || S f ||D
 Euclidean Convexity

Nehari (1976):
f (D) convex  || S f ||D  2
 Spherical Convexity

Mejía, Pommerenke (2000):
f (D) convex  || S f ||D  2
 Hyperbolic Convexity

Mejía, Pommerenke Conjecture (2000):
f (D) convex  || S f ||D  2.3836
Verification of M/P Conjecture
 "A Sharp Bound on the Schwarzian Derivatives
of Hyperbolically Convex Functions,"
Proceeding of the London Mathematical Society
93 (2006), 395_417, with Roger W. Barnard,
Leah Cole and G. Brock Williams.
 "The Verification of an Inequality," Proceedings
of the International Conference on Geometric
Function Theory, Special Functions and
Applications (ICGFT) (accepted) with Roger W.
Barnard.
Special Function Estimates
 Parameter  
 /2
K ( y)
where y  cos 
Special Function Estimates
 Upper bound
Special Function Estimates
 Upper bound
 Partial Sums
Special Function Estimates
Verification
where c  cos 2 ,  
 /2
, 1  x  1
K (cos  )
Verification c3  0
Straightforward to show that q p  0
In qm make a change of variable c  2 y  1
2
Verification c3  0
Obtain a lower bound for qm by estimating 
via an upper bound
*
q
Sturm sequence argument shows m  qm   p8
is non-negative
Grid Estimates
Grid Estimates
 Given


A) grid step size h
B) global bound M for maximum of | f ( x) |
 Theorem Let f be defined on [a, b]. Let
M  max | f ( x) | . Let   0 and suppose that N is
x[ a ,b ]
choosen so that h  (b  a) / N   . Let L be the
f ( x)
lattice L  {a  jh : 0  j  N } . Let m  min
xL
If
 
m  M  ,
2
then f is non-negative on [a, b].
Grid Estimates
 Maximum descent argument
Grid Estimates
 Two-Dimensional Version
Grid Estimates
 Maximum descent argument
Verification
where c  cos 2 ,  
 /2
, 1  x  1
K (cos  )
Verification c2
 The problem was that the coefficient c2 ( , x) was
not globally positive, specifically, it was not
positive for 1  x   4 5 , 0     2 .
 We showed that p4 (t )  0 by showing that
q(t )  0 where
q(t )  c3 ( , x)t 2  c2 ( , x)t  c1 ( , t )
0 < t < 1/4.
Verification c2
 Used Lemma 3.3 to show that the endpoints
q0* (0)  e0 ( y, w) and q0* ( 1 4) are non-negative. We
partition the parameter space into subregions:
Verification c2
 Application of Lemma 3.3 to q0* ( 1 )
4
 After another change of variable, we needed to
show that r  0 where
for 0 < w < 1, 0 < m < 1
Verification c2
Verification c2
[0,1]  [0,1]  [0,1/ 2]  [0,1/ 2]  [0,1/ 2]  [1/ 2,1]
[1/ 2,1]  [0,1/ 2]  [1/ 2,1]  [1/ 2,1]
Quarter Square [0,1/2]x[0,1/2]
M  max{M w , M m }  35
M w  21, M m  35
Grid 50 x 50
m  min r ( w j , mk )  0.400
( w j , mk )L
M   0.350
Question
 Given a function f on an interval [a, b], what
does it take to show that f is non-negative on
[a, b]?
Conclusions
 There are “proof by picture” hazards
 There is a role for CAS in analysis


CAS numerical computations are rational number
calculations
CAS “special function” numerical calculations are
inherently finite approximations
 There are various useful, practical strategies for
rigorously establishing analytic inequalities