Functions of Matrices-Theory and Computation(slides)

CHOL EIG FOR LONG PROD SAVE TRIU
CHOP ELSE FUN LU QR SCHU USER
CLEA END HESS MACR RAND SHOR WHAT
COND EPS HILB MAGI RANK SEMI WHIL
CONJ EXEC IF NORM RCON SIN WHO
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Classic MATLAB
< M A T L A B > Version of 01/10/84 HELP is available <> help Type HELP followed by INTRO (To get started) NEWS (recent revisions) ABS ANS ATAN BASE CHAR DET DIAG DIAR DISP EDIT EXP EYE FILE FLOP FLPS INV KRON LINE LOAD LOG ORTH PINV PLOT POLY PRIN REAL RETU RREF ROOT ROUN SQRT STOP SUM SVD TRIL < > ( ) = . , ; \ / ’ + - * :
COS EXIT IMAG ONES RAT SIZE WHY
MIMS
Nick Higham
Functions of a Matrix
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Classic MATLAB
<> help fun FUN For matrix arguments X , the functions SIN, COS, ATAN, SQRT, LOG, EXP and X**p are computed using eigenvalues D and eigenvectors V . If <V,D> = EIG(X) then f(X) = V*f(D)/V . This method may give inaccurate results if V is badly conditioned. Some idea of the accuracy can be obtained by comparing X**1 with X . For vector arguments, the function is applied to each component.
MIM Matrix
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Interpolation
Definition (Sylvester, 1883; Buchheim, 1886) Distinct e’vals λ1 , . . . , λs , ni = max size of Jordan blocks for λi . Then f (A) = r (A), where r is unique Hermite interpolating poly of degree < s i =1 ni satisfying r (j ) (λi ) = f (j ) (λi ), j = 0 : n i − 1, 2 1 i = 1 : s.
MIMS
Nick Higham
Functions of a Matrix
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Jordan Canonical Form
Z −1 AZ = J = diag(J1 , . . . , Jp ), Jk
mk × mk
=

λk
1 λk
... ...
1 λk

Definition f (A) = Zf (J )Z −1 = Z diag(f (Jk ))Z −1 , f (mk −1) )(λk ) ′ f (λk ) f (λk ) . . . (mk − 1)! . ... . f (Jk ) = . f (λk ) . ... f ′ (λ )
k
f (λk )
MIMS
Nick Higham
Functions of a Matrix
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The Formula for f (Jk )
Write Jk = λk I + Ek ∈ Cmk ×mk . For mk = 3 we have 0 0 1 0 1 0 3 2 = 0. = 0 0 0 , Ek Ek = 0 0 1 , E k 0 0 0 0 0 0 Assume f has Taylor expansion f (j ) (λk )(t − λk )j f (t ) = f (λk ) + f (λk )(t − λk ) + · · · + + ··· . j!
MIMS
Nick Higham
Functions of a Matrix
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Equivalence of Definitions
Theorem The three definitions are equivalent, modulo analyticity assumption for Cauchy. Interpolation: for basic properties. JCF: for solving matrix equations (e.g., X 2 = A, eX = A). For evaluation (normal A). Cauchy: various uses.
MIMS
Nick Higham
ρ(A) < 1.
Functions of a Matrix
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Multiplicity of Definitions
There have been proposed in the literature since 1880 eight distinct definitions of a matric function, by Weyr, Sylvester and Buchheim, Giorgi, Cartan, Fantappiè, Cipolla, Schwerdtfeger and Richter. — R. F. Rinehart, The Equivalence of Definitions of a Matric Function, Amer. Math. Monthly (1955)
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A1/2 = r (A) =
Nick Higham
Functions of a Matrix
Cauchy Integral Theorem
Definition f (A) = 1 2π i
Γ
f (z )(zI − A)−1 dz ,
where f is analytic on and inside a closed contour Γ that encloses λ(A).
Functions of a Matrix: Theory and Computation
Nick Higham School of Mathematics The University of Manchester higham@ /~higham/ Landscapes in Mathematical Science, University of Bath, November 24 2006
Function of a Matrix
f : Cn×n → Cn×n for an underlying scalar function f . These are not matrix functions: trace(A), det(A). The adjugate (or adjoint) matrix. Transfer function f (t ) = B (tI − A)−1 C . sin A = (sin aij ). These are matrix functions: A2 + ···. eA = I + A + 2! A2 A3 + + ···, log(I + A) = A − 2 3 A−1 , A1/2 .
Outline Definition of f (A) Motivation and MATLAB eA and its Frechét derivative A1/2: Modified Newton Methods
1
2
3
4
MIMS
Nick Higham
Functions of a Matrix
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−1
y (0) = y0 ,
′ y ′ (0) = y0
√ ′ sin( A t )y0 ,
√ where A is any square root of A. MATLAB has expm, logm, sqrtm, funm.
MIMS
Nick Higham
Functions of a Matrix
MIMS
Nick Higham
Functions of a Matrix
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Equivalence of Definitions
Theorem The three definitions are equivalent, modulo analyticity assumption for Cauchy. Interpolation: for basic properties. JCF: for solving matrix equations (e.g., X 2 = A, eX = A). For evaluation (normal A). Cauchy: various uses. For computation: Use the definitions (with care). Schur decomposition for general f . Methods specific to particular f .