Mistake and typoes in Backward Substitution, Solving Ax=b, 15_LA_gaussian

[yeejoh]

In Backward Substitution, Solving Ax=b, 15_LA_gaussian, it writes:

Backwards substitution requires us to move from the last row of $U$ and move upwards. We can consider again the general $i$th row with $$ U_{i,i} xi + U{i,i-1} x{i-1} + \ldots + U{i,m-1} x{m-1} + U{i,m} x_m = y_i $$ noting that we are using the fact that the matrix $L$ has 1 on its diagonal. We can now solve for $y_i$ as $$ xi = \frac{1}{U{i,i}} \left( yi - ( U{i,i-1} x{i-1} + \ldots + U{i,m-1} x{m-1} + U{i,m} x_m) \right ) $$

1. Both equations have index mistake ($U{i,i-1} x{i-1}$), which should be $$U_{i,i} xi + U{i,i+1} x{i+1} + \ldots + U{i,m-1} x{m-1} + U{i,m} x_m = y_i$$, $$xi = \frac{1}{U{i,i}} \left( yi - ( U{i,i+1} x{i+1} + \ldots + U{i,m-1} x{m-1} + U{i,m} x_m) \right )$$.
2. Matrix $L$ has nothing to do with backwards substitution. Noting that the matrix $L$ has 1 on its diagonal is unnecessary.
3. The second sentence of ones between two equations should be

We can now solve for $x_i$ as ...

[mandli]

Thanks, (1) has been fixed and (2) and (3) was clearly a copy-paste error (I should have proof read).

[mandli]

This has been addressed now in commit https://github.com/mandli/intro-numerical-methods/commit/74378de11c3146b1999e55a8f6a68768dd0113f9#diff-7cf2daec2ce73ce6343d2c99819e2a16