1. Systems of Linear Equations
Daniel Baur ETH Zurich, Institut für Chemie- und Bioingenieurwissenschaften ETH Hönggerberg / HCI F128 – Zürich
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

2. Problem Definition
We want to solve the problem where x is the vector of unknowns, while A and b are given
Assumptions
The number of equations is equal to the number of unknowns, therefore A is a square matrix
All components of A, b and x are real
A solution exists and it is unique
A-1 exists
A is not singular
A’s columns are linearly independent
A’s rows are linearly independent
det(A) is non-zero
rank(A) is equal to n
Ax = 0 only if x is a null vector
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

3. Analytical Approach
Cramer’s rule (1750): The solution is given by where Ai is defined as follows
b replaces the ith column
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

4. Calculation of the Determinant
The Laplace formula (1772) allows the computation of the determinant of a square matrix:
where Ci,j is the determinant of the sub-matrix obtained by removing the ith row and the jth column of the matrix, multiplied by (-1)i+j:
no jth column
no ith row
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

5. A First Numerical Approach: Gauss Elimination Method
The following operations do not change the result:
Multiply a line by a constant
Substitute a line with a linear combination of multiple lines
Permute the order of lines
This can be used to produce a triangular matrix, which allows the solution to be found easily by substitution
Example system
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

6. Gauss Elimination Example
Triangular System
Multiply by -3
Sum it to 1st line
Multiply by -4
Sum it to 2nd line
Multiply by -3
Sum it to 1st line
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

7. Generalization of the Gauss Elimination Method
We want to find a general procedure to replace one entry in the matrix with a zero; for this we define a multiplier l21
Note that
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

8. The Gauss Elimination in Matrix Notation
If we put all multipliers for one column in one matrix, we get where
This way, a triangular matrix is easily obtained
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

9. Example: Gauss Elimination in Matrix Notation
Total number of operations required
(n-1)(n-2) operations (flops)
n(n-1) operations (flops)
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

10. Gauss Elimination / Transformation Method
The Gauss elimination method is relatively easy to implement (even by hand), but has some distinct disadvantages; namely it
Changes the matrix A
Requires and changes the coefficient vector b
Must be rerun if the vector b changes
If we consider the Gauss method in matrix form on the other hand, we can see that we can use to transform A and b; M is therefore called the Gauss transformation matrix
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

11. The Gauss Transformation Method
We have so far
The final matrix A, which is an right triangular matrix
The matrix M, which is a left triangular matrix
The inverse of M is also a left triangular matrix
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

12. LR (LU) Factorization
We define our (right triangular) solution matrix as follows
If we multiply with L = M-1 from the left on both sides
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

13. LR Factorization (Continued)
The starting matrix A is transformed (factorized) as
If we apply this to a linear equation system, we get
This approach rids us of the disadvantages discussed earlier, because
For every vector b, two simple triangular systems must be solved without factorizing again
The matrices L and R can be stored using the elements of A
If A is modified, it is often possible to modify L and R accordingly without re-factorizing
Note that Gauss elimination is still needed once to compute L and R
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

14. Pivoting
It is easy to see from the definition of the multiplication factors, that the diagonal elements at each step (the pivot values) cannot be equal to zero
This is circumvented by reordering the rows of the matrix A by multiplication with a permutation matrix P
This approach is referred to as LRP-factorization (or LR-factorization with partial pivoting)
Example: a11 = 0  switch the lines  x1 = x2 = 1
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

15. Pivoting (Continued)
Pivoting must also take into account scaling problems; Let us consider a similar example
Pivot elements should have large absolute values
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

16. How does Matlab do it?
When using left divide \, Matlab chooses a procedure depending on the properties of the problem, i.e. if A is
Diagonal, x is computed directly by division
Sparse, square and banded, then banded solvers are used; Either Gauss elimination without pivoting or LR factorization
Left or right triangular, then backsubstitution is used
A permutation of a triangular matrix, it is permuted and 3. applies
Symmetric or Hermitian, then a Cholesky factorization is attempted (A = RR*, where R* is the conjugate transpose of R); If it fails, another indefinite symmetric factorization is attempted
Square but 1 through 5 do not apply, then LR factorization with partial pivoting is applied
Not square, then Householder reflections are used to compute a factorization which leads to a least-squares solution, i.e. a vector x which minimizes the length of Ax – b
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

17. Assignment 1
Find online the template for an LR-factorization of a matrix with partial pivoting. Make the function operational by adding the lines that calculate the M matrix of the current step and the new A matrix.
Why does the transformation matrix T appear in the formula for M? Explain by comparing to the definition of M in the non-pivoting case.
Use the function to factorize the following matrix
Test if the factorization worked, i.e. if LR = A.
Is L in the form you would expect it to be? What implications does this have for its application in solving a linear system (see slide 13) and how could you correct for it?
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

18. Hints
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

19. Assignment 2
Create a random matrix A with dimensions 4x4 and a random column vector b with size 4x1. Solve the system Ax = b.
Create a function that computes the determinant of square matrices using the Laplace formula. Use a recursive approach (see hints).
Use this function to compute the solution of the linear system above using Cramer’s rule.
Do the same for linear equation systems with sizes ranging from 5 to 9.
Read out the CPU time required to solve all these systems with both methods (Cramer’s method and A\b) and compare them.
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems

20. Hints
If you remember the Laplace formula as a sum you can see that the calculation of the determinant requires the calculation of a determinant. You could let the function call itself to do that (recursive function). Remember that the determinant of a 1x1 matrix is equal to its only element.
There are several Matlab commands to read out timings, however the most reliable one is
t_start = tic; statements; t_elapsed = toc(t_start);
Where the time elapsed (in second) is stored in t_elapsed
Daniel Baur / Numerical Methods for Chemical Engineers / Linear Equation Systems