Remarks on Back Substitution
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In literature, back-substitution means:
(1) The system is upper triangular;
(2) There are zero free variables;
(3) The process is carried out from last equation to first
equation, that is, with variables in reversed order.
The textbook definition [EP 3.2] includes free variables, and
therefore it joins several ideas into one algorithm.
WARNING:
For infinitely many solutions, the EP 3.2 "Algorithm Back
Substitution" is slow and error-prone. Recommended is to find the
Last Frame (rref) by back-substitution in the nonzero equations, and
then apply the Last Frame Algorithm.
Technically, back-substitution employs the Toolkit "combo" to
variables in reverse order. The substitution of one equation into
another is exactly the "combo" tool. The catch: A triangular frame
must be found to start back-substitution.
The textbook observes that frames already in echelon form, or
triangular form, can produce faster solutions, provided the order
of the variable list is reversed in the process. Faster
operations result from using the maximum number of zeros in a dot
product. Dot product is the engine in COMBO.
Back-substitution in the textbook [EP 3.2, Algorithm Back
Substitution] includes the last frame algorithm.