Unit 19
Computational Method
Class 12 Mathematics
System of Linear Equations
Class 12 Mathematics – System of Linear Equations Notes PDF
The exact Google Drive ID is not exposed by the current Nepal eNotes page output, so no Drive ID has been invented.
Open Original Nepal eNotes PDF SourceNEB / CDC Focus
This unit covers computational solution of systems of linear equations using Gauss elimination and Gauss-Seidel iteration, together with recognition of ill-conditioned systems.
1. System of Linear Equations
A system of linear equations is a collection of linear equations in the same unknowns that must be satisfied simultaneously.
In matrix form: AX=B.
2. Gauss Elimination Method
Gauss elimination transforms the augmented matrix into upper triangular/echelon form using elementary row operations, then uses back substitution.
- Interchange two rows.
- Multiply a row by a non-zero constant.
- Add a multiple of one row to another row.
- Choose a non-zero pivot in the first column.
- Eliminate entries below the pivot.
- Repeat for the next columns.
- Obtain triangular form.
- Back-substitute from the last equation.
3. Pivoting and Back Substitution
If a pivot is zero or very small, interchange rows where appropriate. A better pivot can reduce numerical error.
After elimination suppose:
2x+y−z=8
−2y+z=−5
−z=−2
Then z=2, y=7/2, and substitute into the first equation for x.
4. Gauss-Seidel Iterative Method
Gauss-Seidel rewrites each equation to solve for one unknown and repeatedly updates values using the newest available estimates.
Starting from guesses x₁⁽⁰⁾,x₂⁽⁰⁾,…, compute successive iterates until changes are sufficiently small.
5. Convergence and Diagonal Dominance
A common sufficient condition for reliable Gauss-Seidel convergence is strict diagonal dominance:
Rearranging equations can sometimes improve diagonal dominance.
6. Ill-Conditioned Systems
An ill-conditioned system is one in which small changes in coefficients or constants may cause large changes in the computed solution.
Warning signs include nearly dependent equations, very small pivots, and strong sensitivity to rounding.
7. Worked Examples
Solve x+y=5, 2x−y=1.
Add the equations: 3x=6⇒x=2; then y=3. The same operations are represented as row elimination.
For 10x+y=11, x+10y=11:
x=(11−y)/10, y=(11−x)/10.
Starting at (0,0), successive updates approach (1,1).
8. Common Mistakes
Apply each operation to every entry, including the RHS.
Interchange rows rather than dividing by zero.
Use the newest values as soon as they are available.
Do not stop solely after a fixed small number of iterations unless instructed.
9. Important Exam Questions
- Solve a 3×3 system by Gauss elimination.
- Explain forward elimination and back substitution.
- Perform Gauss-Seidel iterations to a stated accuracy.
- Check diagonal dominance.
- Explain an ill-conditioned system and its computational effect.
10. One-Minute Revision
- Write the system as an augmented matrix.
- Gauss elimination: triangularize then back-substitute.
- Gauss-Seidel: iterate using newest values.
- Diagonal dominance is a useful convergence condition.
- Ill-conditioned systems are highly sensitive to small data errors.
Discussion
Share a helpful question, idea, or explanation with other students.