ACM Transactions on Mathematical Software (TOMS)

Shampine, L. F.; Baca, L. S.

doi: 10.1145/5960.5964pmid: N/A

Popular codes for the numerical solution of nonstiff ordinary differential equations (ODEs) are based on a (fixed order) Runge-Kutta method, a variable order Adams method, or an extrapolation method. Extrapolation can be viewed as a variable order Runge-Kutta method. It is plausible that variation of order could lead to a much more efficient Runge-Kutta code, but numerical comparisons have been contradictory. We reconcile previous comparisons by exposing differences in testing methodology and incompatibilities of the implementations tested. An experimental Runge-Kutta code is compared to a state-of-the-art extrapolation code. With some qualifications, the extrapolation code shows no advantage. Extrapolation does not appear to be a particularly effective way to vary the order of Runge-Kutta methods. Although an acceptable way to solve nonstiff problems, our tests raise the question as to whether there is any point in pursuing it as a separate method.

Shampine, L. F.; Baca, L. S.

doi: 10.1145/5960.5964pmid: N/A

Popular codes for the numerical solution of nonstiff ordinary differential equations (ODEs) are based on a (fixed order) Runge-Kutta method, a variable order Adams method, or an extrapolation method. Extrapolation can be viewed as a variable order Runge-Kutta method. It is plausible that variation of order could lead to a much more efficient Runge-Kutta code, but numerical comparisons have been contradictory.We reconcile previous comparisons by exposing differences in testing methodology and incompatibilities of the implementations tested. An experimental Runge-Kutta code is compared to a state-of-the-art extrapolation code. With some qualifications, the extrapolation code shows no advantage. Extrapolation does not appear to be a particularly effective way to vary the order of Runge-Kutta methods. Although an acceptable way to solve nonstiff problems, our tests raise the question as to whether there is any point in pursuing it as a separate method.

Lyness, James; Hines, Gwendolen

doi: 10.1145/5960.214318pmid: N/A

Lyness, James; Hines, Gwendolen

doi: 10.1145/5960.214318pmid: N/A

Laub, Alan J.

doi: 10.1145/5960.214319pmid: N/A

Laub, Alan J.

doi: 10.1145/5960.214319pmid: N/A

Deak, I.

doi: 10.1145/5960.214321pmid: N/A

The idea of the economical method is applied for generating samples from any discrete distribution. In the resulting procedure, the expected number of uniformly distributed random numbers is less than in the alias method (practically 1). A refinement gives a version where in limit just one uniformly distributed number is required at the expense of some storage space.

Deak, I.

doi: 10.1145/5960.214321pmid: N/A

The idea of the economical method is applied for generating samples from any discrete distribution. In the resulting procedure, the expected number of uniformly distributed random numbers is less than in the alias method (practically 1). A refinement gives a version where in limit just one uniformly distributed number is required at the expense of some storage space.

Law, Kincho H.; Fenives, Steven J.

doi: 10.1145/5960.5963pmid: N/A

A symbolic node-addition model for matrix factorization of symmetric positive definite matrices is described. In this model, the nodes are added onto the filled graph one at a time. The advantage of the node-addition model is its simplicity and flexibility. The model can be immediately incorporated into finite element analysis programs. The model can also be extended to determine modification patterns in the matrix factors due to changes in the original matrix. For a given matrix K(=LDLt), the time complexity of the algorithm for constructing the structure of the lower triangular matrix factor L is O((L)) where (L) is the number of nonzero entries in L.

Law, Kincho H.; Fenives, Steven J.

doi: 10.1145/5960.5963pmid: N/A