Application of theta functions for numerical evaluation of complete elliptic integrals of the first and second kinds

Technical Report · Wed Mar 01 00:00:00 EST 1989

DOI:https://doi.org/10.2172/6137964· OSTI ID:6137964

Lee, D. K.

An approximation method based on the use of theta functions is shown to be efficient and useful in numerical evaluation of complete elliptic integrals of the first and second kinds, K(k) and E(k), respectively. The integrals are expressed in terms of power series of the form ..sigma..a/sub n/q/sup n2/, 0 less than or equal to q < 1, where q is the nome determined uniquely from a given value of the argument k. The series converse very rapidly except for small domains near /vert bar/k/vert bar/ = 1, where they either converge slowly or fail to converge. When applied on Cray 2 computers for 0 less than or equal to k/sup 2/ less than or equal to 0.9955, the procedure is found to be more efficient than both the Chebyshev approximations of the Hastings form and the standard Gauss arithmetic-geometric mean process. Numerical results that demonstrate the accuracy and efficiency of the approximation method are presented. 8 refs., 8 tabs.

• Research Organization: Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
• DOE Contract Number: AC05-84OR21400
• OSTI ID: 6137964
• Report Number(s): ORNL/TM-11075; ON: DE89010191
• Resource Relation: Other Information: Portions of this document are illegible in microfiche products
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE
FUNCTIONS
NUMERICAL SOLUTION
COMPILED DATA
CRAY COMPUTERS
ELLIPTICAL CONFIGURATION
INTEGRAL EQUATIONS
MAGNETIC FIELDS
COMPUTERS
CONFIGURATION
DATA
EQUATIONS
INFORMATION
NUMERICAL DATA
990230* - Mathematics &amp; Mathematical Models- (1987-1989)
990210 - Supercomputers- (1987-1989)