contour integrals and their applications
DESCRIPTION
CONTOUR INTEGRALS AND THEIR APPLICATIONS. Wayne Lawton Department of Mathematics National University of Singapore S14-04-04, [email protected] http://math.nus.edu.sg/~matwml. ARYABHATA. - PowerPoint PPT PresentationTRANSCRIPT
CONTOUR INTEGRALS AND THEIR APPLICATIONS
Wayne Lawton
Department of Mathematics
National University of Singapore
S14-04-04, [email protected]
http://math.nus.edu.sg/~matwml
ARYABHATAcharacterized the set { (x, y) } of integer solutions of the equation
1byaxwhere a and b are integers. Clearly this equation admits a solution if and only if a and b have no common factors other than 1, -1 (are relatively prime) and then Euclid’s algorithm gives a solution. Furthermore, if (x,y) is a solution then the set of solutions is the infinite set
}integer an is :),( { kkaykbx
Van der Warden, Geometry and Algebra in Ancient Civilizations, Springer-Verlag, New York, 1984.
BEZOUTinvestigated the polynomial version of this equation
12211 QPQP
Bezout identities in general rings arise in numerous areas of mathematics and its application to science and engineering:
Clearly this equation has a solution iff 1P
common roots and then Euclid’s algorithm gives a solution.have no2
Pand
Algebraic Polynomials: control, Quillen-Suslin TheoremLaurent Polynomials: wavelet, splines, Swan’s TheoremH_infinity: the Corona TheoremEntire Functions: distributional solutions of systems of PDE’sMatrix Rings: control, signal processing
E. Bezout, Theorie Generale des Equations Algebriques, Paris, 1769.
INEQUALITY CONSTRAINTSareTheorem If RPLP
21 P,PLP
Proof. Let LP
21 Q,Q 1QPQP 2211 0 on the unit circle T
then andon T
21 B,B Treal on with 1BPBP 2211 1
21 )PP(G
0
)]PP(G1[BG
)]PP(G1[BG
Q
Q
212
211
2
1
Choose a LP that is real on T with
then choose
12121i
121 )PP()]PP(B1][)PP(-[G
W.Lawton & C.Micchelli, Bezout identities with inequality constraints, Vietnam J. Math. 28:2(2000) 97-126
W.Lawton & C.Micchelli, Construction of conjugate quadrature filters with specified zeros, Numerical Algorithms, 14:4 (1997) 383-399
UPPER LENGTH BOUNDS
Theorem
ZZG ),0(:There exists with
}0)(P)(P:|min{| 21
Proof: Uses resultants.
)}}P(),(Pmin{max{L21
)}}Q(),(Qmin{max{B21
),(GB L
Furthermore, for fixed
L),)(5L()L,(: 2/L2562/L5 2
OG and for fixed L
0),log()L,(: 14/-L2
OG
LOWER LENGTH BOUNDS
Theorem
n4B 14
Proof: See VJM paper.
21 P,P and with
For any positive integer n, there exist LP
n4)(P)(P 21
Question: Are there better ways to obtain bounds that ‘bridge the gap’ between the upper and lower bounds
CONTOUR INTEGRALrepresentation for the Bezout identity is given by
Theorem Let are a disjoint contours and the
dzizzk
kk
1
3 )](T)(2[)(P)(B
\Cz
21,k contains all roots of
k,-3P and
21 then forexcludes all roots of
Proof Follows from the residue calculus.
where
interior ofk
1,2kk ,P
are LP, real on T, and satisfy the Bezout identity.
SOLUTION BOUNDS
Lemma
}0{~
)T,(m)~
,(d2
||)P,P,~
(M 21
where
},T:|)(B{|max 1,2kk zza contour that is disjoint fromand whose (annular) interior contains T
},~
:|)(P|{max)P,P,~
(M 1,2kk21 zz
}~
,:||{min)~
,(d }:|)(T|{min)T,(m
)(length||
CONTOUR CONSTRUCTION
)())((P)( 1 zPzz z on T,0Pk
khence if -invariant contours then it suffices are
to consider these quantities inside of the unit disk D.For k=1,2 let kD union of open disks of radius
},min{ 81
4 L centered at zeros of
k-3P in D
Since
and0D be the disk of this radius centered at 0.
kk DE if 0k DD else 0kk DDE D,EF kk D,Fˆ
kk )ˆ(ˆkkk
Theorem 222/29 ))(1( LLL
CONCLUSIONS AND EXTENSIONS
The contour integral method provide sharper bounds for
and therefore for B than the resultant method but
dziz
zTz
kk
k
1)](T)(2[
)(1)P-(n
)()(B
sharper bounds are required to ‘bridge the gap’.
Contour integrals for BI with n > terms are given by
wherek encloses all zeros of T except for those of
kP
Residue current integrals give multivariate versions