2.9 Incomplete Elliptic Integrals

Transcription

1 .9 Inomplete Ellipti Integrals A. Purpose An integral of the form R t, P t) 1/) dt 1) in whih Pt) is a polynomial of the third or fourth degree that has no multiple roots, and R is a rational funtion of t and Pt) 1/, is either elementary, or is an ellipti integral. It is always possible to express integrals of the form of Eq. 1) linearly in terms of elementary funtions and three ellipti integrals of anonial form. These funtions are desribed more ompletely in [1] and []. Several anonial forms have been proposed, but the most widely used are due to Jaobi, Legendre and Carlson. In eah of Eqs. ) 4) we present first Jaobi s and then Legendre s form of the anonial ellipti integrals: F ϕ, k) = = Eϕ, k) = = y ϕ y ϕ 1 t ) 1/ 1 k t ) 1/ dt 1 k sin θ ) 1/ dθ ) 1 t ) 1/ 1 k t ) 1/ dt 1 k sin θ ) 1/ dθ 3) Πϕ, α, k) y = 1 α t ) 1 1 t ) 1/ 1 k t ) 1/ dt = ϕ 1 α sin θ ) 1 1 k sin θ ) 1/ dθ 4) in whih y = sin ϕ. If ϕ is equal to π/, the integrals are said to be omplete, otherwise they are inomplete. Carlson s forms of the anonial ellipti integrals are R D a, b, ) = 3 t + a) 1/ t + b) 1/ t + ) 3/ dt 5) in whih a and b are nonnegative suh that a + b > and is positive; if either a or b is zero, the integral is omplete, otherwise it is inomplete, R F a, b, ) = 1 t + a) 1/ t + b) 1/ t + ) 1/ dt 6) 1997 Calif. Inst. of Tehnology, 15 Math à la Carte, In. in whih a, b and are nonnegative and at most one of them is zero; if one of a, b or is zero, the integral is omplete, otherwise it is inomplete, and R J a, b,, r) = 3 t + r) 1 t + a) 1/ t + b) 1/ t + ) 1/ dt 7) in whih a, b and are nonnegative, and at most one of them is zero, and r is nonzero; if one of a, b or is zero, the integral is omplete, otherwise it is inomplete. Notie that R D a, b, ) = R J a, b,, ). But the neessity to ompute R D a, b, ) arises frequently in pratie, and a proedure espeially tailored to ompute R D a, b, ) is more effiient than omputing R J a, b,, ). The funtion R C a, b) = R F a, b, b) is elementary, but also appears frequently. A proedure is provided to ompute R C a, b). Identify a, b and suh that a b, and assume a <. Then 3/ 3 R D a, b, ) = k sin 3 [F ϕ, k) Eϕ, k)] 8) ϕ 1/ F ϕ, k) R F a, b, ) = 9) sin ϕ 3/ 3 R J a, b,, r) = α sin 3 [Πϕ, α, k) F ϕ, k)] ϕ 1) where os ϕ = a/, k = b)/ a) and α = r)/ a). The subprograms desribed in this hapter evaluate the anonial forms of inomplete ellipti integrals, using either the Legendre or the Carlson parameterization. B. Usage B.1 Program Prototype, Single Preision, Legendre s Form, E and F PHI, K, F, E Assign values to PHI and K. B.1.a CALL SELEFI PHI, K, F, E, ) PHI [in] Argument, ϕ, of the ellipti integral. Require PHI π/. K [in] Modulus, k. Require K 1.. July 11, 15 Inomplete Ellipti Integrals.9 1

2 F [out] Fϕ, k) with ϕ given by PHI and k given by K. E [out] Eϕ, k) with ϕ given by PHI and k given by K. [out] status indiator: = no errors 1 = Magnitude of argument too large, PHI > π/. = Magnitude of Modulus too large, K > = PHI = π/ and K = 1, F is infinite. B. Program Prototype, Single Preision, Legendre s Form, Π PHI, K, ALPHA, PI Assign values to PHI, K and ALPHA. CALL SELPII PHI, K, ALPHA, PI, ) B..a PHI [in] Argument, ϕ, of the ellipti integral. Require PHI π/. K [in] Square of the modulus, k. Require k sin ϕ 1.. See Setion E. ALPHA [in] Charateristi, α. Require α sin ϕ 1.. See Setion E. PI [out] Πϕ, α, k), with ϕ given by PHI, α given by ALPHA and k given by K. [out] status indiator. If =, there were no errors. Other values are produed by proedures SRFVAL and SRJVAL see Setions B.5 and B.6) whih are used in omputing Πφ, α, k). B.3 Program Prototype, Single Preision, Carlson s Form, R C X, Y, RC Assign values to X and Y. B.3.a CALL SRCVAL X, Y, RC, ) X, Y [in] Arguments of the ellipti integral. Require X, Y. See Setion E. RC [out] The omputed value of R C X, Y). [out] Status indiator: = no errors 1 = X <. or Y =.. = X + Y too small See Setion E). 3 = X or Y or X + Y too large See Setion E). 4 = Y < and Y too large and X too small See Setion E). B.4 Program Prototype, Single Preision, Carlson s Form, R D X, Y, Z, RD Assign values to X, Y and Z. B.4.a CALL SRDVAL X, Y, Z, RD, ) X, Y, Z [in] Arguments of the ellipti integral. Require X, Y, X + Y >, Z >. See Setion E. RD [out] The omputed value of R D X, Y, Z). [out] Status indiator: = no errors 1 = X <. or Y <. or Z <.. = X + Y too small or Z too small See Setion E). 3 = X or Y or Z too large See Setion E). B.5 Program Prototype, Single Preision, Carlson s Form, R F X, Y, Z, RF Assign values to X, Y and Z. B.5.a CALL SRFVAL X, Y, Z, RF, ) X, Y, Z [in] Arguments of the ellipti integral. Require X, Y, Z, at most one of X, Y or Z equal zero. See Setion E. RF [out] The omputed value of R F X, Y, Z). [out] Status indiator: = no errors 1 = X <. or Y <. or Z <.. = X + Y or X + Z or Y + Z too small See Setion E). 3 = X or Y or Z too large See Setion E). B.6 Program Prototype, Single Preision, Carlson s Form, R J X, Y, Z, R, RJ Assign values to X, Y, Z and R. CALL SRJVAL X, Y, Z, R, RJ, ).9 Inomplete Ellipti Integrals July 11, 15

3 B.6.a X, Y, Z, R [in] Arguments of the ellipti integral. Require X, Y, Z, at most one of X, Y or Z equal zero, R. See Setion E. RJ [out] The omputed value of R J X, Y, Z, R). [out] Status indiator: = no errors 1 = X <. or Y <. or Z <. or R =.. = X + Y or X + Z or Y + Z or R too small See Setion E). 3 = X or Y or Z or R too large See Setion E). B.7 Modifiations for Double Preision For double preision usage, hange the type statements to DOUBLE PRECISION and hange the subprogram names SELEFI, SELPII, SRCVAL, SRDVAL, SRFVAL and SRJVAL to DELEFI, DELPII, DRCVAL, DRDVAL, DRFVAL and DRJVAL, respetively. π Λ α, β) = Λα, β, π/) [ = sin β R F, os α, 1) 1 ] 3 sin αr D, os α, 1) R F os β, 1 os α sin β, 1 ) 1 3 os α sin 3 β R F, os α, 1) R D os β, 1 os α sin β, 1) 1) The variants of Legendre s integrals used by Bulirsh in [4] and [5] are el1x, k ) = xr F x, 1 + x ), 13) elx, k, a, b) = axr F x, 1 + x ) b a)x3 R D x, 1 + x ) 14) ele3x, k, p) = xr F x, 1 + x ) p)x3 R J x, 1 + x, 1 + px ) 15) C. Examples and Remarks C.1 Related Funtions Logarithms, inverse irular funtions and inverse hyperboli funtions an be expressed in terms of R C, see [9, pp. 163, 186]: ln x)/x 1) = R C x), x), x > ; sin 1 x)/x = R C 1 x, 1), 1 x 1; sinh 1 x)/x = R C 1 + x, 1), < x < ; os 1 x)/1 x ) 1 = R C x, 1), x 1; osh 1 x)/x 1) 1 = R C x, 1), x 1; tan 1 x)/x = R C 1, 1 + x ), < x < ; tanh 1 x)/x = R C 1, 1 x ), 1 < x < 1; ot 1 x = R C x, x + 1), x < ; oth 1 x = R C x, x 1), x > 1. The first seven of these allow omputing nearly indeterminate forms with more auray than would be possible using the naïve formulation. Heuman s lambda funtion [3] is a variant of Legendre s third integral: elk, p, a, b) = ar F, k, 1) = 1 3 b pa)r J, k, 1, p ) C. Whih Proedure Should Be Used? 16) Several fators influene the hoie of proedure. If one needs to write a simple program and use it one, one should probably hoose the proedure that evaluates the funtions in the form most similar to the way the problem is posed. If one needs to write a program that will have substantial use, one should usually prefer SE- LEFI to SRDVAL and SRFVAL, as the former is up to 3 times faster than the latter two. An exeption to this rule ours if one needs to ompute R D a, b, ) with < maxa, b), in whih ase the parameters for SELEFI will be out of range. If auray is an issue but speed is not, one may prefer SRDVAL and SRFVAL to SELEFI, at least for omputing Fϕ, k). See testing in Setion D below). SRCVAL is somewhat slower, on an IBM PC/AT with a 1 os α sin β ) numeri data proessor, than using the equivalent Fortran intrinsi funtions. This is no surprise, as most 1/ os Λα, β, ϕ) of the intrinsi funtions are implemented by hardware. α sin β os β But the inverse hyperboli funtions are not. SRCVAL = sin ϕ R F os ϕ, 1 sin α sin ϕ, 1) is roughly the same speed as the proedures in Chapter.1. As mentioned above, it may be advantageous to R J os ϕ, use SRCVAL to ompute nearly indeterminate forms. sin α sin 3 ϕ os α sin β ) 1 sin α sin ϕ, 1, 1 sin α sin ϕ ) SELPII is implemented by using SRJVAL and SRFVAL 1 os α sin 11) see Eqs. 9) and 1) above). Thus, there is no speial β July 11, 15 Inomplete Ellipti Integrals.9 3

4 advantage in speed or auray to one or the other. The sole riterion is how losely the forms of the funtions evaluated diretly by the proedures math the forms of the funtions the user needs to evaluate. D. Funtional Desription D.1 Properties of the Funtions The first form given in Eqs. ) 4) is the Jaobi or algebrai form. When expressed in this form Eq. ) is finite for all real and omplex y, inluding, has a simple pole of order 1 for y =, and is logarithmially infinite for y = 1/α. D. Method of Computation The proedure SELEFI is based upon a proedure ELLPI developed by Allan V. Hershey and modified by Alfred H. Morris, desribed in [6]. The proedure uses series expansions due to DiDonato and Hershey, desribed in [7]. The proedure SELPII is based upon a proedure EPI developed by Alfred H. Morris, desribed in [5]. It omputes Πϕ, k, α ) using Eqs. 9) and 1), as omputed by SRFVAL and SRJVAL. The proedures SRCVAL, SRDVAL, SRFVAL and SRJVAL are based on proedures developed by B. C. Carlson and Elaine M. Notis, desribed in [8] and [9]. All of the referened proedures were revised to be onsistent with low level modules and naming onventions of MATH77. D.3 Testing The single preision programs for Eϕ, k), Fϕ, k), R D a, b, ) and R F a, b, ) were tested on an IBM PC/AT using IEEE arithmeti) by omparison to double preision results, as desribed below. The relative preision of IEEE single preision arithmeti is ρ = The auray of proedure SELEFI was assessed by omparing its results to double preision results obtained by applying Eqs. 8) and 9), with R D a, b, ) and R F a, b, ) evaluated by DRDVAL and DRFVAL, respetively. The auray of proedures SRDVAL and SRFVAL was assessed by omparing their results to double preision results obtained by applying Eqs. 8) and 9), with Eϕ, k) and Fϕ, k) evaluated by DELEFI. To test SELEFI, the retangular region ϕ π/ k 1 of the ϕ k plane was divided into regions, and a point was randomly seleted in eah region. To test SRDVAL and SRFVAL, the argument was set to 1., the retangular region a < 1 b < 1 of the a b plane was divided into regions, and a point was randomly seleted in eah region. The maximum relative and absolute errors are summarized in the following table. Max. Rel. Max. Abs. Funtion Error Error Eϕ, k).8ρ.98ρ Fϕ, k) 5.4ρ 15.91ρ R D a, b, 1) 1.ρ 3.1ρ R F a, b, 1) 1.35ρ.55ρ Errors in Fϕ, k) inrease as the arguments approah the infinite singularity at ϕ = π/ and k = 1. Referenes 1. Milton Abramowitz and Irene A. Stegun, Handbook of Mathematial Funtions, Applied Mathematis Series 55, National Bureau of Standards 1966) Chapter 17, Paul F. Byrd and Morris D. Friedman, Handbook of Ellipti Integrals for Engineers and Sientists, Springer Verlag, Berlin 1971). 3. H. Kuki, Tables of omplete ellipti integrals, J. Math. and Physis 1941) Roland Bulirsh, Numerial alulation of ellipti integrals and ellipti funtions, Numerishe Mathematik ) Roland Bulirsh, Numerial alulation of ellipti integrals and ellipti funtions, Numerishe Mathematik ) Alfred. H. Morris, Jr., NSWC Library of Mathematis Subroutines. Tehnial Report NSWCDD/TR-9/45, Naval Surfae Warfare Center, Dahlgren, VA USA Jan. 1993) Armido R. DiDonato and Allan V. Hershey, New formulas for omputing inomplete ellipti integrals of the first and seond kind, J. ACM ) B. C. Carlson, Computing ellipti integrals by dupliation, Numerishe Mathematik ) B. C. Carlson and Elaine M. Notis, Algorithm 577: Algorithms for inomplete ellipti integrals [S1], ACM Trans. on Math. Software 7, 3 Sept. 1981) B. C. Carlson, Speial Funtions of Applied Mathematis, Aademi Press, New York 1977). E. Error Proedures and Restritions The proedure SELEFI requires ϕ π/, and k 1. Proedure SELPII omputes Πϕ, k, α ) from R J a, b,, r) and R F a, b, ) using Eqs. 9) and 1). The initial values for the arguments are a = os ϕ, b = 1 k sin ϕ, r = 1 α sin ϕ, and = maxa, b, r). Then a, b and r are replaed by a, b and r, respetively. SELPII requires ϕ π/. Restritions on k and α are enfored indiretly by restritions on.9 4 Inomplete Ellipti Integrals July 11, 15

5 a, b and imposed by SRFVAL and SRJVAL, desribed below. The ranges for ϕ and k an be extended using formulae 113.1, 113., 114.1, 115.1, 115., 16., 161. and 16. from [], or formulae through from [1]. Denote the largest representable magnitude by Ω, and the smallest nonzero representable magnitude by ω. General restritions on the arguments to proedures SR- CVAL, SRDVAL, SRFVAL and SRJVAL were desribed above in Setion B. SRCVAL requires X + Y 5ω, X Ω/5, Y Ω/5, and, if Y <.36/ ω it requires X ωω) /5. Denote the mahine round-off level by ρ, that is, ρ is the smallest positive number suh that the representation of 1 + ρ is different from 1. Let ε be the solution of the equation ρ = 3ε 6 1 ε) 3/, Ω D = Ω /3 and ω D = εω /3 /1. SRDVAL requires X + Y ω D and Z ω D, X Ω D, Y Ω D and Z Ω D. SRFVAL requires X + Y 5ω, X + Z 5ω, Y + Z 5ω, X Ω/5, Y Ω/5 and Z Ω/5. Let Ω J = Ω/5) 1/3 /5 and ω J = 5ω) 1/3. SRJVAL requires X + Y ω J, Y + Z ω J, X + Z ω J, R ω J, X Ω J, Y Ω J, Z Ω J and R Ω J. The aessible ranges of the arguments may be extended beyond the ranges admissible in the proedures by using the homogeneity of the funtions: R F ka, kb, k) = k 1/ R F a, b, ), R J ka, kb, k, kr) = k 3/ R J a, b,, r). and If any of the restritions above is violated, all proedures return an error indiator in the argument named, and invoke the error message proessor see Chapter 19.) with LEVEL =. The proedure ERMSET see Chapter 19.) may be used to affet the default error proessing ation. F. Supporting Information The soure language for these subroutines is ANSI Fortran 77. The proedures SELEFI and SELPII were written by W. V. Snyder in Deember 199, based on earlier proedures desribed by Alfred H. Morris, Naval Surfae Warfare Center, Dahlgren, VA in [5]. The proedures SRCVAL, SRDVAL, SRFVAL and SRJVAL were written by W. V. Snyder in Deember 199, based on earlier proedures desribed by Carlson and Notis in [9]. Entry Required Files DELEFI AMACH, DELEFI, DERM1, DERV1, DLNREL, ERFIN, ERMSG DELPII AMACH, DELPII, DERM1, DERV1, DRCVAL, DRFVAL, DRJVAL, ERFIN, ERMSG DRCVAL AMACH, DERM1, DERV1, DRCVAL, ERFIN, ERMSG DRDVAL AMACH, DERM1, DERV1, DRDVAL, ERFIN, ERMSG DRFVAL AMACH, DERM1, DERV1, DRFVAL, ERFIN, ERMSG DRJVAL AMACH, DERM1, DERV1, DRCVAL, DRFVAL, DRJVAL, ERFIN, ERMSG SELEFI AMACH, ERFIN, ERMSG, SELEFI, SERM1, SERV1, SLNREL SELPII AMACH, ERFIN, ERMSG, SELPII, SERM1, SERV1, SRCVAL, SRFVAL, SRJVAL SRCVAL AMACH, ERFIN, ERMSG, SERM1, SERV1, SRCVAL SRDVAL AMACH, ERFIN, ERMSG, SERM1, SERV1, SRDVAL SRFVAL AMACH, ERFIN, ERMSG, SERM1, SERV1, SRFVAL SRJVAL AMACH, ERFIN, ERMSG, SERM1, SERV1, SRCVAL, SRFVAL, SRJVAL July 11, 15 Inomplete Ellipti Integrals.9 5

6 DRSELI program DRSELI >> DRSELI Krogh Changes t o use M77CON >> DRSELI WV Snyder Create s e p a r a t e s i n g l e and d o u b l e demos. >> DRSELI WV Snyder JPL O r i g i n a l ode. S r e p l a e s? : DR?ELI,?RCVAL,? ELEFI,? ELPII,?RDVAL,?RFVAL,?RJVAL Demonstration d r i v e r f o r inomplete e l l i p t i i n t e g r a l proedures. real ALPHA, E, F, K, K, PHI, PI, R, RC, RD, RF, RJ real SINPHI, T, U, X, Y, Z integer Compute ar s i n e x using ASIN and RC, f o r x =.5 print, I d e n t i t i e s from write up : x =. 5 e a l l s r v a l 1. e x x, 1. e, r, i e r r ) i f i e r r. eq. ) then t = asin x ) x r print ASIN. 5 ). 5 RC1.5,1) =, g15. 8 ), t else print SRCVAL r e t u r n s e r r o r s i g n a l, i 1 ), i e r r Evaluate i d e n t i t i e s g i v e n by e q u a t i o n s 8 1) in t h e write up with k = 1/, s i n phi ) = 1/4, alpha = 1/, = 1. From t h i s, we have a = 3/4, b = r = 7/8. alpha =. 5 e k = sqrt. 5 e ) k =. 5 e s i n p h i =. 5 e phi = asin s i n p h i ) r =.875 e x =. 7 5 e y =.875 e z = 1. e a l l s e l e f i phi, k, f, e, i e r r ) i f i e r r. ne. ) then print SELEFI r e t u r n s e r r o r s i g n a l, i 1 ), i e r r a l l s e l p i i phi, k, alpha, pi, i e r r ) i f i e r r. ne. ) then print SELPII r e t u r n s e r r o r s i g n a l, i 1 ), i e r r a l l s r d v a l x, y, z, rd, i e r r ) i f i e r r. ne. ) then print SRDVAL r e t u r n s e r r o r s i g n a l, i 1 ), i e r r a l l s r f v a l x, y, z, r f, i e r r ) i f i e r r. ne. ) then print SRFVAL r e t u r n s e r r o r s i g n a l, i 1 ), i e r r.9 6 Inomplete Ellipti Integrals July 11, 15

7 a l l s r j v a l x, y, z, r, r j, i e r r ) i f i e r r. ne. ) then print SRJVAL r e t u r n s e r r o r s i g n a l, i 1 ), i e r r u = sqrt z 3) rd t = 3. e / k s i n p h i 3) f e ) r = u t ) / u print Equation 8 ), LHS RHS)/LHS =, g15. 8 ), r u = sqrt z ) r f t = f / s i n p h i r = u t ) / u print Equation 9 ), LHS RHS)/LHS =, g15. 8 ), r u = sqrt z 3) r j t = 3 / alpha s i n p h i 3) p i f ) r = u t ) / u print Equation 1 ), LHS RHS)/LHS =, g15. 8 ), r 99 stop end ODSELI I d e n t i t i e s from write up : ASIN. 5 ). 5 RC1.5,1) = E 7 Equation 8 ), LHS RHS)/LHS = E 5 Equation 9 ), LHS RHS)/LHS =. Equation 1 ), LHS RHS)/LHS = E 5 July 11, 15 Inomplete Ellipti Integrals.9 7