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Manuscript Title: HYMNIA - band matrix package for solving eigenvalue problems.
Authors: R. Gruber
Program title: HYMNIA
Catalogue identifier: ACWC_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 10(1975)30
Programming language: Fortran.
Computer: CDC CYBER 7326.
Operating system: SCOPE 3.4.2.
RAM: 18K words
Word size: 60
Peripherals: disc.
Keywords: General purpose, Numerical mathematics, Eigenvalue problem, Inverse iteration, Band matrix, Hermitian matrix.
Classification: 4.8.

Nature of problem:
Stability problems in physics or engineering lead to eigenvalue problems of the type Ax = lambda Bx, where B is positive definite and A can either be real symmetric of hermitian. Both matrices have a band structure.

Solution method:
HYMNIA is a band matrix package which solves the above mentioned eigenvalue problem by the method of simultaneous inverse vector iteration. Only the eigenvalues with lowest absolute value are obtained directly, but with an eigenvalue shift lambdaO it is possible to get any of them. The advantage of HYMNIA, which is also the name of the CDC main program that tests the two main subprograms <SIVI> and <CSIVI>, is that all operations are performed with in the half bandwidth m of the one- dimensionally stored matrices A and B or order n. By decomposing the matrices (~nm**2 operations before the iteration, only a number of point operations proportional to nm has to be performed for the iteration part. The NEG facility gives the number of negative eigenvalues for the shifted problem and thus enables us to know which eigenvalue has been calculated. <SIVI> not only solves the eigenvalue problem, but can also be used simply to decompose A and/or B. The cases of singular A and non- positive B are detected and diagnosed. Convergence information is given by the parameters NCONV and CONV.

The matrix B has to be real symmetric and positive definite. The matrix A can either be real symmetric <SIVI> or hermitian <CSIVI>. Both, A and B have to have a band structure.

Unusual features:
The test program HYMNIA reads input data using NAMELIST, which is not STANDARD FORTRAN but is available on most computers. Another non- standard utility is the CDC random generator RANF called by the subprograms <SIVI>, <CSIVI>, <ORNOS> and <CORNOS>.