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Change of variables is an operation that is related to substitution. However these are different operations, as can be seen when considering differentiation or integration (integration by substitution). A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth-degree polynomial:
The second part (sometimes called back substitution) continues to use row operations until the solution is found; in other words, it puts the matrix into reduced row echelon form. Another point of view, which turns out to be very useful to analyze the algorithm, is that row reduction produces a matrix decomposition of the original matrix.
The shuffling method is a very common form of data obfuscation. It is similar to the substitution method but it derives the substitution set from the same column of data that is being masked. In very simple terms, the data is randomly shuffled within the column. [3]
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The lower bound of multiplications needed is 2mn+2n−m−2 (multiplication of n×m-matrices with m×n-matrices using the substitution method, m⩾n⩾3), which means n=3 case requires at least 19 multiplications and n=4 at least 34. [41] For n=2 optimal 7 multiplications 15 additions are minimal, compared to only 4 additions for 8 multiplications.
Forward and back substitution [ edit ] A matrix equation in the form L x = b {\displaystyle L\mathbf {x} =\mathbf {b} } or U x = b {\displaystyle U\mathbf {x} =\mathbf {b} } is very easy to solve by an iterative process called forward substitution for lower triangular matrices and analogously back substitution for upper triangular matrices.
Often, theory can establish the existence of a change of variables, although the formula itself cannot be explicitly stated. For an integrable Hamiltonian system of dimension , with ˙ = / and ˙ = /, there exist integrals .
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