EXTENSIONS OF PAD´ E-TYPE APPROXIMANTS Ernst Joachim Weniger Theoretical Chemistry University of Regensburg, Germany joachim.weniger@chemie.uni-regensburg.de SC2011 International Conference Scientific Computing on S. Margherita di Pula, Sardinia, Italy October 10-14, 2011 (with corrections) 1
Glory and Misery of Power Series ⊲ Power series are the most important tools of calculus. ⊲ They possess highly advantageous analytical features. ⇒ They are used abundantly in all applications of mathematics. ⊲ From a purely numerical point of view a power series expansion is a mixed blessing. ⊲ Power series converge in their circles of con- vergence, which may shrink to a single point, and they diverge outside of their circles of convergence. ⇒ With the exception of a few fortunate cases like ∞ z n /n ! � exp( z ) = n =0 power series are not suited to evaluate a func- tion effectively and reliably for all z ∈ C . 2
Rational Functions ⊲ Rational functions are ratios of two polyno- mials P and Q of degrees m, n ∈ N 0 : � m µ =0 p µ z µ P m ( z ) = ( z − α 1 ) · · · ( z − α m ) Q n ( z ) = ( z − β 1 ) · · · ( z − β n ) . � n ν =0 q ν z ν ⊲ Power series and rational functions have largely complementary properties: • Power series expansions for numerous func- tions are explicitly known. • For few functions only, rational approxi- mants are explicitly known. • The construction of a finite set of ratio- nal approximants to a given function is usually a nontrivial numerical problem. 3
Advantages of Rational Functions ⊲ Rational approximants normally have (much) better numerical properties than the power series from which they are derived. ⊲ A function, whose power series has a nonzero , but finite radius of convergence must have at least one singularity on the boundary of the circle of convergence. ⇒ Rational approximants can approximate func- tions outside the circles of convergence of their power series. ⊲ Numerical problems can only occur in the im- mediate vicinity of the poles of the rational function. ⊲ But even the poles of a rational approximant – the zeros of its denominator polynomial – can provide useful information about the function. ⊲ Rational functions can simulate the cut of a function. 4
Pad´ e Approximants ⊲ Formal power series for f : C → C : ∞ γ n z n . � f ( z ) = n =0 ⊲ Pad´ e approximant to f : [ m/n ] f ( z ) = P [ m/n ] ( z ) , m, n ∈ N 0 , Q [ m/n ] ( z ) P [ m/n ] ( z ) = p 0 + p 1 z + · · · + p m z m , Q [ m/n ] ( z ) = q 0 + q 1 z + · · · + q n z n . ⊲ Analyticity of [ m/n ] f ( z ) at z = 0 is guaran- teed by the Baker condition q 0 = 1. ⇒ The remaining m + n + 1 polynomial coeffi- cients p 0 , p 1 , . . . , p m and q 1 , q 2 , . . . , q n are de- termined by requiring that the modified accu- racy-through-order relationship Q [ m/n ] ( z ) f ( z ) − P [ m/n ] ( z ) = O z m + n +1 � � holds as z → 0. 5
⇒ The accuracy-through-order relationship leads to a system of m + n +1 coupled linear equa- tions. ⊲ The denominator coefficients q 1 , q 2 , . . . , q n are determined via the n equations min( m + ν,n ) � q κ γ m + ν − κ = 0 , 1 ≤ ν ≤ n , κ =0 ⊲ The numerator coefficients p 0 , p 1 , . . . , p m are determined via the m + 1 equations min( µ,n ) � q κ γ µ − κ = p µ , 0 ≤ µ ≤ m , κ =0 which correspond to the m + 1 leading terms of a Cauchy product of two power series. ⇒ The difficult part is the computation of the denominator coefficients q 1 , q 2 , . . . , q n . ⊲ These equations show that the coefficients γ 0 , γ 1 , . . . , γ m + n of the power series for f are needed for the construction of [ m/n ] f ( z ). 6
Pad´ e-Type Approximants: Heuristic Mo- tivation ⊲ The polynomial coefficients of a Pad´ e ap- proximant [ m/n ] f ( z ) can be computed if the numerical values of the power series coeffi- cients γ 0 , γ 1 , . . . γ m + n are known. ⊲ No further information about the function f ( z ), which is to be approximated, is needed. ⊲ This feature is also shared by other compu- tational schemes for Pad´ e approximants such as Wynn’s celebrated epsilon algorithm. ⊲ This feature of Pad´ e approximants is highly advantageous, and it contributed significantly to their usefulness and popularity. ⊲ But this advantage can also become a dis- advantage. We often have some knowledge about the function, which we want to ap- proximate. ⊲ Unfortunately, there is no obvious way of uti- lizing such an information in the case of Pad´ e approximants. 7
Pad´ e-Type Approximants: Theory ⊲ In 1979, Brezinski introduced his so-called e-type approximant Pad´ ( m/n ) f ( z ) = U ( m/n ) ( z ) , m, n ∈ N 0 , V ( m/n ) ( z ) U ( m/n ) ( z ) = u 0 + u 1 z + · · · + u m z m , V ( m/n ) ( z ) = v 0 + v 1 z + · · · + v n z n , which look like Pad´ e approximants [ m/n ] f ( z ). Their theory was fully developed in a mono- graph by Brezinski (1980). ⊲ However, it is now assumed that the denomi- nator polynomial V ( m/n ) ( z ) is explicitly known. ⇒ The coefficients of the numerator polyno- mial are determined via the via the modified asymptotic condition ( z → 0) V ( m/n ) ( z ) f ( z ) − U ( m/n ) ( z ) = O( z m +1 ) . ⇒ Explicit expression as a Cauchy product: min( µ,n ) m U ( m/n ) ( z ) = z µ � � v ν γ µ − ν . µ =0 ν =0 8
Cauchy Products of Power Series ⊲ Assume ∞ n ϕ n z n , ϕ ν z ν , � � Φ( z ) = Φ n ( z ) = n =0 ν =0 ∞ n ψ n z n . ψ ν z ν . � � Ψ( z ) = Ψ n ( z ) = n =0 ν =0 ⊲ Standard form of the Cauchy product with truncation: ∞ n z n � � Φ( z ) Ψ( z ) = ϕ n − ν ψ ν n =0 ν =0 N n z n � � = ϕ ν ψ n − ν n =0 ν =0 + O( z N +1 ) , z → 0 . ⊲ Alternative Cauchy product involving partial sums Ψ n ( z ): N ϕ ν z ν Ψ N − ν ( z ) � Φ( z ) Ψ( z ) = ν =0 + O( z N +1 ) , z → 0 . 9
Alternative Expression for Pad´ e-Type Ap- proximants ⊲ In the theory of sequence transformations, it is frequently more convenient to use the partial sums n γ ν z ν � f n ( z ) = n ∈ N 0 . ν =0 of a power series as input data and not their coefficients γ ν . ⇒ Alternative explicit expression for the numer- ator polynomial min( m,n ) v µ z µ f m − µ ( z ) U ( m/n ) ( z ) = � µ =0 and the Pad´ e-type approximant � min( m,n ) v µ z µ f m − µ ( z ) µ =0 ( m/n ) f ( z ) = . � n ν =0 v ν z ν ⊲ The alternative expression for U ( m/n ) ( z ) can also be used for the numerator polynomial P [ m/n ] ( z ) of a Pad´ e approximant [ m/n ] f ( z ). 10
Pad´ e-Type Approximants: General Con- siderations ⊲ The computation of the numerator polyno- mials U ( m/n ) ( z ) of a Pad´ e-type approximant ( m/n ) f ( z ) is fairly simple once the denomina- tor polynomial V ( m/n ) ( z ) is explicitly known. ⇒ The real challenge is the choice of a suitable class of denominator polynomials if we want to employ Pad´ e-type approximants in conver- gence acceleration and summation processes. ⊲ The poles of a function are only rarely known. A fortunate counter-example is the digamma function: ∞ ψ ( z ) = − γ − 1 z � z + k ( k + z ) k =1 z � = 0 , − 1 , − 2 , · · · . A Pad´ e-Type approximant to ψ ( z ) was con- structed by Weniger (2003). ⇒ In general, alternative approaches for the de- termination of the denominators are neces- sary. 11
Levin’s Transformations ⊲ In 1973, Levin introduced a sequence trans- formation which utilizes the information con- tained in explicit estimates of the truncation error of the input sequence ⇒ The explicit utilization of additional informa- tion makes Levin’s transformation potentially very powerful. ⊲ Levin’s idea was later extended in articles by Levin and Sidi (1981), Sidi (1979), Sidi and Levin (1982), Weniger (1989, 2004), and Homeier (2000). ⇒ Many new, more or less closely related se- quence transformation were derived which also utilize the information contained in explicit truncation error estimates. ⊲ It can be shown that certain variants of Levin- type transformations are actually Pad´ e-type approximants [Weniger 2004]. 12
Levin-Type Transformations ⊲ The Levin-type transformations considered in Weniger (2004) all possess the general structure ( s n , ω n ) = ∆ k [ P k − 1 ( n ) s n /ω n ] T ( n ) k ∆ k P k − 1 ( n ) /ω n ] k � P k − 1 ( n + j ) s n + j ( − 1) j � k � j ω n + j j =0 = . k � P k − 1 ( n + j ) ( − 1) j � k � j ω n + j j =0 P k − 1 ( n ) is a polynomial of degree k − 1 in n , and the { ω n } ∞ n =0 are remainder estimates. ⊲ Obviously, ∆ k { P k − 1 ( n ) /ω n } � = 0 must hold for all finite k, n ∈ N 0 . ⊲ The { ω n } ∞ n =0 should reproduce the leading order asymptotics of the actual remainders { r n } ∞ n =0 : � � �� r n = s n − s = ω n c + O 1 /n , n → ∞ . 13
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