Vector spherical harmonics on @math have wide applications in geophysics, quantum mechanics and astrophysics. In the representation of a tangent field, one needs to evaluate the expansion and the Fourier coefficients of vector spherical harmonics. In this paper, we develop fast algorithms (FaVeST) for vector spherical harmonic transforms for these evaluations. The forward FaVeST which evaluates the Fourier coefficients has computational steps proportional to @math for @math number of evaluation points. The adjoint FaVeST which evaluates a linear combination of vector spherical harmonics with degree up to @math for @math evaluation points is proportional to @math . Numerical examples illustrate the accuracy and efficiency of FaVeST.
We present in this paper a spectrally accurate numerical method for computing the spherical vector spherical harmonic expansion of a function vector field with given (elemental) nodal values on a spherical surface. Built upon suitable analytic formulas for dealing with the involved highly oscillatory integrands, the method is robust for high mode expansions. We apply the numerical method to the simulation of three-dimensional acoustic and electromagnetic multiple scattering problems. Various numerical evidences show that the high accuracy can be achieved within reasonable computational time. This also paves the way for spectral-element discretization of 3D scattering problems reduced by spherical transparent boundary conditions based on the Dirichlet-to-Neumann map. A memory package for storing data and capable of being added to or replaced while being provided with a battery backup in computer systems which can be mounted with a plurality of said memory packages. A memory package also has a function to reduce power consumption of the battery backup to a minimum, when the memory package with battery backup is inserted into the computer system without the main power being applied. Consequently, even if additional memory packages are inserted in computer systems installable with memory packages having battery backups, only a weak current need be supplied from the backup power supply so that the battery will not run down after a short time. The memory packages can therefore be added and replaced while still storing data internally and the memory capacity can be changed. Also, a memory system is provided including a power supply supplying power supply lines connected to the memory packages. Diode and or surge protection arrangements are also provided to facilitate hot line insertion and removal.