Row Reduction Demo

This linear algebra tool performs Gaussian elimination, while generating an explanation of every step. For rational inputs, the answer is exact, in contrast with most numerical software.

Comma or space can be used to separate entries in a row.
Semicolon or newline can be used to separate rows.
The input syntax allows Lisp-style comments:
- The part of a line following ';' is a comment.
- Everything between comment delimiters '#|' and |#' is a comment, e.g. '#| Hello, world! |#'.
Please make sure rows have the same number of entries.
Rational numbers (fractions) are allowed as coefficients, for example 35/29 Fractions and integers can have an arbitrary number of digits.
Floating point numbers can be single-precision or double precision. A single-precision number is entered as 1.23456f3 (=1.23456*10^3) A double-precision number is entered as 1.2345678901234d7 (=1.2345678901234*10^7) If no exponent part is specified, double precision is assumed.
Complex coefficients are allowed. For example, 1+2i is entered either in Lisp format #C(1 2) or in the traditional format 1+2i, 1+2*i, 1+2I, or 1+2j, or 1+2J. The real and imaginary parts may be integer, fractional or floating point.
NOTE: When reporting basic and free variables, it is assumed that the matrix is the augmented coefficient matrix and that the last column contains the right-hand sides, and thus does not have a variable associated with it. If the matrix is the coefficient matrix (of a homogenous system, for example), one has to take into account that the last variable may be basic or free, depending on whether there is a pivot in it or not.
NOTE: The program uses a heuristic guess regarding entries of the matrix which would be zero, should there not be round off error. This is how this works. If the program encounters only entries with magnitude smaller than the value of pivot threshold in some column, all entries rejected in this column as pivots are zeroed out and search for next pivot continues in the next column. One can easily suppress the heuristic by setting the thershold to 0. The default value of 1e-6 allows to solve many small examples accurately, which would fail due to round off. An example is the augmented matrix:
```
	    [ 1 1.000001 1 ; 
              0.999999 1 0.999995]
	  
```
Without the heuristic, the program would find a pivot in the last column and declared the system inconsistent. With the heuristic, the program finds that the system has rank 1 (1 basic variable) and finds this reduced row echelon form:
```
	    [ 1 1.000001 1 ; 
              0 1        0 ]
	  
```
This is most likely what we want. In one case, when the entries are rational numbers (fractions), one should set the pivot threshold to 0 if the program starts replacing non-zero entries with zeros (unlikely for typical "book" examples).

Should row scaling be performed after finding the pivot?
Should decimals be converted to rationals?
Should back-substitution be performed?
Should partial pivoting strategy be used?
Should rows be divided first by their maximum elements?
Treat pivot candidates with magnitude smaller than this value as zero:
Rendering document as
Output complex numbers in format A+Bi?

Automatic Row Reduction Input Form

Options