lin_eq_with_sql

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A method in SQL to solve linear equations

Raymond Juillerat

March 1, 2008

Abstract

This article is typically an internet paper. It is published a first time and will be completed later

step by step. This little document presents a method implemented fully with MySql for solving

linear equations.

Foreword

For a rapid estimation of some approximations regarding their validity, it may be interesting to solve

a set of linear equations directly within the same environnement used to produce the basis matrix, i.

e. a SQL environnement. Freeware doing this ist rather seldom to be found on the internet.

Therefore this solution, implemented and tested in a MySql environnement. Such a solution was

no realisable before stored procedures were implemented. Compared with other implementations,

like with Oracle, there are some drawbacks, as an example, MySql doesnt allow a declaration of an

inner procedure in a procedure. However, a migration to another db-system should be rather easy.

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SQL MySql Solution of linear equations

1 Some features of the implementation

1.1 Stability of the estimations

This implementation is based on the Householder transform, which is known as beeing stable. Its

requests more resources than a Gauss elimination, but offers against this drawback the advantage of

a greater stability.

1.2 Flexibility of the method

The method can be used to solve a set of linear equations of any order. Only the execution time

will differ following a cubic increase related to the order. This is a great advantage of a SQL-based

solution compared with an algorithmic based one, coded in Pascal, for example.

2 Technical aspects

2.1 Description of the matrix

The equations are to be given in a table that describes the matrix. This entity is very simple, the

attributes beeing column, row and coefficient. Column and row are integer values, the coefficients

are double floating numbers. The values for attribute row are from 0 to (order-1). The values for

attribute column are from 0 to order, the greatest one for the constants, meant as situated right to the

equal sign.

The table is a representation of the equations

anixn + a(n1)ixn1 + ...+ a2ix2 + a1ix1 = consti

n1

i=0

where the row is equal to i and the column is equal to 0 for ani and n for consti.

2.2 The table of results

The results for the variables are brought to a vector. The table bearing this results is an entity com-

posed of the attributes row and coefficient. Its also used for the reflection vector necessary to perform

the Householder transform.

2.3 About the package

The package[2] contains 4 files, the first one (create_hh_tables.sql) containing the code creating the

matrix and the vector that have to be defined in a first step. The second file (hh.sql) contains the stored

procedures as they has been used with MySql. The third file (hh_t.sql) and the last one (hh_simu.sql)

contain each a small example.

2.3.1 Please dont reinvent the wheel

Its clear that the necessary explanations exist [1] without which this work was not possible. That

paper gives a full explanation of the Householder algorithm and a solution written in Pascal. In the

published codefor SQL a great part of the variables names are derived from that work, the SQL

variable v_wii from the Pascal variable wii for example. The routinesnames are also derived from

the Pascal proceduresnames. In the Housholder transformation the table-name hh_re_vect is the

image of the mirroring vector vii. After the transformation it is used to put the result of the final

calculation on it.

[email protected] 2 english version


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SQL MySql Solution of linear equations

References

[1] http://brinch-hansen.net/papers/1992a.pdf : Explanations of the Housholder transform with an

implemention in Pascal P. Brinch Hansen, Householder reduction of linear equations, ACMComputing Surveys 24, 2 (June 1992), 185-194. Copyright c1992, Association for Computing

Machinery, Inc.

[2] http://www.raym.org/sql/hh_package.zip

[email protected] 3 english version

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