HP Prime:
Gauss-Jordan Elimination Method
I received an email requesting some programs of various
numerical methods. One of the methods
is the Gauss-Jordan Elimination Method.
Basically, the Gauss-Jordan Elimination Method is a
step-by-step method of matrix row operations to reduce a matrix A = [ X | Y ]
where X is (mostly) a square component joined by a column vector Y to the form
[ I | R ], which I represents an identity matrix portion where the diagonal
elements are 1.
You can quickly execute the method by use of the RREF
(Reduced Row Echelon Form) function, which is present on many graphing
calculators (if not all of them these days).
The HP Prime program GAUSSJORDAN is shows a step by step
method. What the program does is:
1. Take the dimensions of the matrix.
2. Starting with column 1, pivot on element (1,1). This is accomplished by one of two
operations. SCALEADD which multiplies
the first row by a factor and adds it to a target row k. The goal is reduce the element (k,1) to 0,
for all k ≠ 1.
3. Once step 2 is completed, the SCALE command is used to
divide the value of element (1,1) to reduce it to 1, if necessary.
4. Repeat steps 2 and 3 for each column until the number
the rows is reached.
Notes:
1. HP Prime’s SCALEADD is the *Row+ command for the Casio
and TI graphing calculators. Similarly, HP
Prime’s SCALE is the *Row command for the Casio and TI graphing
calculators.
2. The program requires that the matrix have all elements
that are (1,1), (2,2), (3,3), etc. are non-zero. Otherwise an error occurs. You can swap rows by the rowSwap command if
necessary to get the matrix in proper form.
3. While the program GAUSSJORDAN shows you step by step, it
shows one order of approach, which may or may not be the most efficient number
of steps.
4. If the HP Prime
is in Standard mode, you may see 0.9999999999999 or some number to the 10^-13
power. This is due to the rounding
mechanisms. Feel free to round these
numbers to 1 and 0, respectively.
With all this in mind, here is the program:
Program GAUSSJORDAN
EXPORT GAUSSJORDAN(mat)
BEGIN
// Guass-Jordan Elimination
// 2015-12-04
// Matrix MUST have
// mat[k,k]≠0
LOCAL l,r,c,j,k,h,v;
PRINT();
l:=SIZE(mat);
r:=l[1]; c:=l[2];
j:=1;
// Main row loop
FOR k FROM 1 TO r DO
// Secondary row loop
FOR h FROM 1 TO r DO
// k = target row
// h = test row
// Pivot operation
IF h≠k THEN
v:=−mat[h,k]/mat[k,k];
mat:=SCALEADD(mat,v,k,h);
PRINT();
PRINT("After Step "+j);
PRINT(mat);
WAIT(0);
j:=j+1;
END;
IF mat[k,k]≠1 THEN
v:=1/mat[k,k];
mat:=SCALE(mat,v,k);
PRINT();
PRINT("After Step "+j);
PRINT(mat);
WAIT(0);
j:=j+1;
END;
END;
END;
PRINT();
PRINT("Result:");
PRINT(mat);
// Final WAIT not needed
RETURN mat;
END;
Examples:
M2 = [[1,2,4,1] [3, -3, 3, 0] [6, 6, 8, 5]]
Final Result* (see Note 4) of GAUSSJORDAN(M2):
[[1,0,0,0.53333333333]. [0,1,0,0.43333333333], [0,0,1,-0.1]]
M3 = [[0,4,-1,-1], [2,5,2,2], [3,3,6,3]].
Running GAUSSJORDAN(M3) with M3 as is will get an
error. Why? See M3[1,1] = 0. We must make it non-zero. Do this by swapping rows.
M3≔rowSwap(M3,1,2) to make the matrix [[2,5,2,2],
[0,4,-1,-1], [3,3,6,3]]. Now you are
ready to go.
GAUSSJORDAN(M3) returns [[1,0,0,2.6], [0,1,0,-0.4],
[0,0,1,-0.6]]
Thank you Luis for the email and the suggestions.
Source: Cazelais,
Gilles. “Gauss-Jordan Elimination Method”
http://pages.pacificcoast.net/~cazelais/251/gauss-jordan.pdf Retrieved December 4, 2015
Until next time, stay safe everyone because it is a crazy
planet we live on.
Eddie
