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diagonal matrix. It is also possible matrix with zeros below the main diagonal as follows. [A,b]?[D,e] This still leaves some work to be done in order to solve the simplified equiv tem $Dx = e$ for $x$, but the additional computational effort is not significant lowing algorithm is an implementation of the partial elimination approach Gaussian elimination method. Like Alg. 2.2.1, it returns an error code $E$ that the coefficient matrix $A$ is singular. Gaussian Elimination 1. Set $E = 1$ and form the $n \times (n + 1)$ augmented matrix $C = [A, b]$. 2. For $j = 1$ to $n$ do (a) Compute the pivot index $j \le p \le n$ such that: $|C_{pj}| = \max_{j \le i \le n} |C_{ij}| (b) If $C_{pj} = 0$, set $E = 0$ and exit. (c) If $p > j$ interchange rows $p$ and $j$. (d) For each $i > j$, subtract $C_{ij}/C_{pj}$ times row $j$ from row $i$. 3. Partition $C$ as $C = [D, e]$ where $D$ is $n \times n$ and $e$ is $n \times 1$. 4. For $j = n$ down to $1$ compute $x_j = \frac{1}{D_{jj}} \left( e_j - \sum_{i=j+1}^n D_{ij} x_i \right)$ Alg. 2.3.1 is a two-pass algorithm. The first pass is the forward eliminatic n steps (1) through (3) where the first $n$ columns of $C$ are row-reduced to

          diagonal matrix. It is also possible
matrix with zeros below the main diagonal as follows.
[A,b]?[D,e]
This still leaves some work to be done in order to solve the simplified equiv
tem $Dx = e$ for $x$, but the additional computational effort is not significant
lowing algorithm is an implementation of the partial elimination approach
Gaussian elimination method. Like Alg. 2.2.1, it returns an error code $E$ that
the coefficient matrix $A$ is singular.
Gaussian Elimination
1. Set $E = 1$ and form the $n \times (n + 1)$ augmented matrix $C = [A, b]$.
2. For $j = 1$ to $n$ do
(a) Compute the pivot index $j \le p \le n$ such that:
$|C_{pj}| = \max_{j \le i \le n} |C_{ij}|
(b) If $C_{pj} = 0$, set $E = 0$ and exit.
(c) If $p > j$ interchange rows $p$ and $j$.
(d) For each $i > j$, subtract $C_{ij}/C_{pj}$ times row $j$ from row $i$.
3. Partition $C$ as $C = [D, e]$ where $D$ is $n \times n$ and $e$ is $n \times 1$.
4. For $j = n$ down to $1$ compute
$x_j = \frac{1}{D_{jj}} \left( e_j - \sum_{i=j+1}^n D_{ij} x_i \right)$
Alg. 2.3.1 is a two-pass algorithm. The first pass is the forward eliminatic
n steps (1) through (3) where the first $n$ columns of $C$ are row-reduced to

diagonal matrix. It is also possible
matrix with zeros below the main diagonal as follows.
[A,b]?[D,e]
This still leaves some work to be done in order to solve the simplified equiv
tem Dx = e for x, but the additional computational effort is not significant
lowing algorithm is an implementation of the partial elimination approach
Gaussian elimination method. Like Alg. 2.2.1, it returns an error code E that
the coefficient matrix A is singular.
Gaussian Elimination
1. Set E = 1 and form the n × (n + 1) augmented matrix C = [A, b].
2. For j = 1 to n do
(a) Compute the pivot index j ≤ p ≤ n such that:
|Cpj| = maxj ≤ i ≤ n |Cij|
(b) IfCpj = 0, setE = 0and exit.
(c) Ifp > jinterchange rowspandj.
(d) For eachi > j, subtractCij/Cpjtimes rowjfrom rowi.
3. PartitionCasC = [D, e]whereDisn ×nandeisn ×1.
4. Forj = ndown to1computexj = (1)/(Djj) ( ej - ∑i=j+1^n Dij xi )Alg. 2.3.1 is a two-pass algorithm. The first pass is the forward eliminatic
n steps (1) through (3) where the firstncolumns ofCare row-reduced to

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