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Scientific Subroutine Package¶

ND 210007/ND 210009

Compute¶

(a) ( \binom{8}{5} ), (b) ( \binom{9}{7} )¶

[ \binom{n}{r} = \frac{n(n-1)(n-2)\ldots}{(n-1)(n-2)\ldots r} ]

[ \begin{align} (a) \quad \binom{8}{5} &= \frac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} = 56 \ &= \frac{6 \cdot 16!}{6! \cdot 11!} + \frac{11 \cdot 16!}{6! \cdot 11! \cdot 10!} = \ldots \ &= \frac{17 \cdot 16!}{6! \cdot 11!} = \frac{17!}{6! \cdot 11!} \end{align} ]

Further Calculations¶

[ \frac{27 \cdot 26 \cdot 25}{25!} = \frac{2}{2} ]

[ \frac{11}{13 \cdot 12} = \frac{11!}{14 \cdot 13 \cdot 12 \cdot 1} ]

Compute: ( \frac{13!}{11!} ), ( \frac{7!}{10!} )¶

stateDiagram-v2
    state "q00" as q00
    state "q01" as q01
    state "q10" as q10
    state "q11" as q11
    state "q12" as q12

    q00 --> q01: a
    q01 --> q11: a
    q01 --> q12: a
    q10 --> q11: a
    q10 --> q12: b
    q11 --> q00: b
    q11 --> q10: b
    q12 --> q10: b
    q12 --> q11: a

Photo representation of a flowchart containing state and transition actions related to computations and subroutines.

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SCIENTIFIC SUBROUTINE PACKAGE¶

INTRODUCTION¶

The Scientific Subroutine Package comprises a set of subroutines written in ND FORTRAN IV for users of ND Computer Systems. It is primarily developed to furnish the mathematical tools required to simplify the solution of complex problems encountered in engineering and the sciences. The Scientific Subroutine package — SSP — relieves the sophisticated user from the arduous and mundane task of program development and testing. For the novice, it eliminates the chore of acquiring a detailed knowledge of the novel and esoteric field of numerical mathematics and iterative algorithms. Special care has been taken in the selection of the various algorithms to ensure that truncation errors, numerical instability etc. will not jeopardize the final result. Execution speed, storage economy and ease of use have also been major concerns. However, the imperative objective has been to produce a program package which generates valid and reliable results.

FEATURES¶

Modular design
Input/output independent
Programmed in ND FORTRAN IV
Can be combined in any sequence to solve complex problems
Uniformly documented with examples of more complex subroutines
Covers most mathematical requirements in statistics, research and engineering
Parameter transfer by FORTRAN Call statement from user program

PRODUCT DESCRIPTION¶

The ND 10007 — 48 bit format and the ND 10009 — 32 bit format Scientific Subroutine Package — SSP — consists of a collection of I/O independent, modular subroutines written in ND FORTRAN IV. The subroutines can be grouped into three categories: Statistics, matrix manipulation and "other" mathematics for use in science, industry and engineering.

In the development of the SSP, special concern has been given such factors as execution speed, storage economy and past experience with the applied algorithms.

The SSP provides the user with a wide collection of I/O independent computational building blocks that can be freely combined with the user's input and arranged to form the sequence required to solve a particular problem. As no fixed maximum dimensions are defined within the subroutines, no restriction is imposed on the user with respect to the size of the data arrays to be processed.

Many of the matrix manipulation subroutines handle symmetric and diagonal matrices (stored in compressed format to improve storage economy) as well as general matrices. This can result in considerable savings in data storage for large arrays. The use of more complex subroutines (or groups of them) is illustrated in the program documentation by sample main programs with I/O. All subroutines are documented uniformly.

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SCIENTIFIC SUBROUTINE PACKAGE¶

Individual subroutines, or a combination of them, can be used to carry out the listed functions in the following areas:

Statistics¶

Analysis of variance (factorial design)
Correlation analysis
Multiple linear regression
Polynomial regression
Canonical correlation
Factor analysis (principal components, varimax)
Discriminant analysis
Time series analysis
Data screening and analysis
Nonparametric tests
Random number generation (uniform, normal)

Matrix Manipulation¶

Inversion
Eigenvalues and vectors (real symmetric case)
Simultaneous linear algebraic equations
Transpositions
Matrix arithmetic (addition, product, etc.)
Partitioning
Tabulation and sorting of rows or columns
Elementary operations on rows or columns

Other Mathematical Areas¶

Integration of given or tabulated functions
Integration of first-order differential equations
Fourier analysis of given or tabulated functions
Bessel and modified Bessel function evaluation
Gamma function evaluation
Legendre function evaluation
Elliptic, exponential, sine, cosine, Fresnel integrals
Finding real roots of a given function
Finding real and complex roots of a real polynomial
Polynomial arithmetic (addition, division, etc.)
Polynomial evaluation, integration, differentiation

REQUIREMENTS¶

Memory requirements will depend on which subroutine(s) is used.

REFERENCES¶

Scientific Subroutine Package Application
ND-60.037 EN Description