Example of Gaussian Elimination; Matrix Math in Basic 7.0/7.1 (51605)

   MatSEqnS% (for single-precision)
   MatSEqnD% (for double-precision)
   MatSEqnC% (for currency data type)
				

   QBX /L MATBEFR.QLB
				

MORE INFORMATION

Definitions

1. A matrix is a two-dimensional array in Basic.
2. A vector is a one-dimensional array in Basic.
3. An identity matrix is a square array composed of 1's along the diagonal from upper left to lower right, with all else 0's (zeros).

Gaussian Elimination

A linear equation of n variables (unknowns) has the following form:

   a1*x1 + a2*x2 + an*xn = b
				

where: Linear equations do not involve any products or roots of variables. All variables are to the first power, and don't appear as arguments of trigonometric, logarithmic, or exponential functions.

A solution of a linear equation is a sequence of n numbers (s1 through sn) such that the equation is satisfied when we substitute x1=s1, x2=s2, ..., xn=sn.

A set of multiple linear equations in the variables x1 through xn is called a system of linear equations. The set (vector) of constants s1 through sn is called a solution of the system if it provides a solution for every equation in the system. Every system of linear equations has either no solutions, exactly one solution, or infinitely many solutions.

A system of m linear equations in n unknowns can be written as follows in Basic:

   a(1,1)*x1 + a(1,2)*x2 + ... + a(1,n)*xn = b(1)

   a(2,1)*x1 + a(2,2)*x2 + ... + a(2,n)*xn = b(2)

      ...                  ...

   a(m,1)*x1 + a(m,2)*x2 + ... + a(m,n)*xn = b(m)
				

If you mentally keep track of the location of the +'s, the x's, and the ='s, the arrays a(m,n) and b(m) provide a shorthand notation for the system of linear equations. In elementary linear algebra texts, a(m,n) and b(m) together make what is called the "augmented matrix."

Again, our goal is to discover the unknown values s1 through sn that, when assigned to variables x1 through xn, solve every equation.

Gaussian elimination reduces the augmented matrix [the combination of a(m,n) and b(m)] to a matrix of reduced row-echelon form, which looks like a square identity matrix attached to a 1 by m vector [ b() ]. The vector b() contains the solution set (s1 through sn) of the system of linear equations.

The Gaussian elimination functions MatSEqnS%, MatSEqnD%, and MatSEqnC% accept a square matrix a() and a vector b() as input arguments (together composing the input-augmented matrix), and give the solution in the one-dimensional array b(). After you invoke the function, a() is replaced with the identity matrix, and the solution values overwrite the input arguments that you had placed in b(). The n solution values in b(), when assigned to variables x1 through xn, satisfy every equation in the system.

For more information about linear algebra, the following is an excellent text:

Code Example

' This program demonstrates Gaussian elimination to solve a set of
' linear equations using double-precision variables.
' The MATSEQND, MATSEQNS, and MATSEQNC matrix math routines are
' provided in the Matrix Math Toolbox in Microsoft Basic 7.00 for
' MS-DOS and MS OS/2.
'
' To run this program in the QuickBasic Extended editor, load the
' MATBEFR.QLB Quick library as follows:
'
'     QBX /L MATBEFR.QLB
'
' MATFEFR.QLB and the related .LIB files are documented in the INCLUDE
' file 'MATB.BI' but NOT in "Microsoft Basic 7.0: Language Reference."

DECLARE FUNCTION MatSEqnD% (A() AS DOUBLE, b() AS DOUBLE)

' The above DECLARE statement is all that is needed from the following
' include file:   REM $INCLUDE: 'matb.bi'
DEFDBL A-Z
OPTION BASE 1    ' Use OPTION BASE 1 for easier reference.
DIM A(3, 3), b(3)

' The following system of linear equations:
PRINT "       x +   y + 2*z = 9"
PRINT "     2*x + 4*y - 3*z = 1"
PRINT "     3*x + 6*y - 5*z = 0"
' ...can be represented in the following matrix:
'     1  1  2  9
'     2  4 -3  1
'     3  6 -5  0
' ...which can be placed in arrays a() and b() respectively as follows:
'     a(1,1)  a(1,2)  a(1,3)  b(1)
'     a(2,1)  a(2,2)  a(2,3)  b(2)
'     a(3,1)  a(3,2)  a(3,3)  b(3)
A(1, 1) = 1
A(1, 2) = 1
A(1, 3) = 2
A(2, 1) = 2
A(2, 2) = 4
A(2, 3) = -3
A(3, 1) = 3
A(3, 2) = 6
A(3, 3) = -5
b(1) = 9
b(2) = 1
b(3) = 0
errcode% = MatSEqnD%(A(), b())
PRINT "The following values for x, y, and z solve all three equations:"
PRINT "x="; b(1)
PRINT "y="; b(2)
PRINT "z="; b(3)

' Here is the output, accurate to within double-precision limits:
'    x= 1.00000000000001
'    y= 2
'    z= 3