Algebraic, transcendental (i.e., involving trigonometric and
exponential functions), ordinary differential equations, or
partial differential equations...
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology
1. Analytical Method – is one that produces either exact
or approximate solutions in closed form
2. Components – the elements of a vector
3. Conformable – matrices with identical dimensions
4. Accuracy – is a measure of the nearness of a value for
the true value
5. Precision – is a measure of the clustering of values
near each other
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology
6. Triangular matrix – a square matrix in which all the
elements on one side of the diagonal are zero
7. Gauss elimination – methods for solving a system;
reducing the matrix to the upper triangular form, and
then back to substitution
8. Double sequence – is a function of domain of
ordered pairs (i, j)of integer and with range consisting
of a portion of the real number system
9. Non- singular matrix – a square matrix with a nonzero determinant
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology
:
:
:
:
:
:
:
:
:
:
Where xj (j=1,2,…m) denotes the unknown variable
aij (i=1,2,…n; j=1,2,…m) denotes the coefficients of
the unknown variable
bi (i=1,2,…n) denotes the non-homogeneous terms
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

One that produces either exact or
approximate solutions in closed form.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
1. A unique solution – a consistent set of solutions
2. No solution – an inconsistent set of equations
3. An infinite number of solutions – a redundant
set of equations
4. The trival solution xj = 0 – a set of
homogenous equations
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
1. Direct methods – are systematic procedures,
based on algebraic elimination, that obtain the solution
in a fixed number of operations.
2. Iterative methods – obtain the solution
asymptotically by an iterative procedure. A trial solution
is assumed, the trial solution is substituted into the
system of equations to determine the mismatch in the
trial solution, and an improved solution is obtained
from the mismatch data.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Definition 1
An (m x n) or (m, n) matrix is a rectangular array
of quantities arranged in m rows and n columns.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Definition 2
A matrix with only one row is a special kind of matrix
known as a row vector.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Definition 3
A matrix with only one column is a special kind of
matrix known as a column vector.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Definition 4
The (n x m) or (n, m) matrix obtained from a given
(m x n) or (m, n) A by interchanging its rows and
columns is called the transpose of A denoted by the
symbol AT.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Definition 5
A square matrix is a matrix where the dimensions
m is equal to n.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Definition 6
A symmetric matrix is one where aij = aji for all i’s
and j’s.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Definition 7
A square matrix in which each element not on the
principal diagonal is zero is called a diagonal matrix.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Definition 8
A square matrix in which every element below the
principal diagonal is zero is said to be upper
triangular matrix.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Definition 9
A square matrix in which every element above the
principal diagonal is zero is said to be lower triangular
matrix.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Definition 10
A square matrix in which all elements equal to zero,
with the exception of a band centered on the main
diagonal is called a bonded matrix (e.g. tridiagonal
matrix).
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Definition 11
A diagonal matrix in which each diagonal element is 1
is called a unit matrix or identity matrix.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Definition 12
A matrix in which every element is zero is called a null
matrix or zero matrix.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Definition 13
The determinant of an (n, n) square matrix A is written
as lAl and is defined by either of
or
in which cij is known as the cofactor of the element aij.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Definition 14
The cofactor cij of an (n, n) square matrix A is
obtained by first removing row i and column j to form
an (n-1, n-1) matrix and then performing the operation
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Definition 15
The augmented matrix is obtained by adjoining the
column vector b to the coefficient matrix A.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Definition 16
A coefficient matrix with a zero determinant is
singular, a unique solution for x requires a nonsingular coefficient matrix.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
A. Methods for Triangular Matrices
It involves reduction of matrix equation into one of
the forms:
, L = lower triangular matrix
, U = upper triangular matrix
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
B. Cramer’s Rule
 Gives the components xi of x in terms of
determinants according to:
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
B. Cramer’s Rule Example
 Use the Cramer’s rule to solve:
0.3x1 + 0.52x2 + x3 = -0.01
0.5x1 + x2 + 1.9x3 = 0.67
0.1x1 + 0.3x2 + 0.5x3 = -0.44
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
C. Gaussian Elimination
 a method for solving a system of the type (A• x = b)
wherein the goal is to reduce it to the upper triangular
form and then use the back substitution scheme to
obtain the components from each of the remaining
equations.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
C. Gaussian Elimination Example
Use Gaussian elimination to solve:
3x1 - 0.1x2 - 0.2x3 = 7.85
0.1x1 + 7x2 - 0.3x3 = -19.3
0.3x1 – 0.2x2 + 10x3 = 71.4
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
D. Gauss-Jordan Method
 a variation of Gauss Elimination wherein the goal is
to reduce the original matrix to a diagonal form.
 not popular since there is neither a reduction in
programming complexity nor increased efficiency.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
D. Gauss-Jordan Example
Use Gauss-Jordan to solve the previous problem:
3x1 - 0.1x2 - 0.2x3 = 7.85
0.1x1 + 7x2 - 0.3x3 = -19.3
0.3x1 – 0.2x2 + 10x3 = 71.4
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
E. LU Decomposition Method
 is another elimination method of solving general
systems of linear algebraic equations wherein the
objective is to find a lower triangular factor L and an
upper triangular factor U such that the system of
equations can be transformed according to
Where A* = matrix after row exchange have been made to
allow the factors L and U to be computed accurately;
b* = vector b after an identical set of row exchanges.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
E. LU Decomposition Example
Solve the previous problem:
3x1 - 0.1x2 - 0.2x3 = 7.85
0.1x1 + 7x2 - 0.3x3 = -19.3
0.3x1 – 0.2x2 + 10x3 = 71.4
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
1. Division by Zero
2. Round-off Errors
3. Ill-Condition Systems
 is one where a small changes in one or more of the
coefficients results in large changes in the solution.
4. Singular Systems
 is worse than ill-conditioned because two equations
in the system are identical.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
1. Pivoting
2. Use of more significant figures
3. Scaling
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
1. Solve the following systems:
x1 + 2x2 = 10
1.1x1 + 2x2 = 10.4
Then solve it again, but with the
coefficient of x1 in the second equation
modified slightly to 1.05.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
2. Evaluate the determinant of the
following systems:
3x1 + 2x2 = 18
-x1 + 2x2 = 2
And solve also the determinant in
prob. 1.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
A. Gauss-Seidel Method
 Iterative or approximate methods .
 Start the process by assigning initial values
(guessing a value) and then use a systematic method
to obtain a refined estimate of the root. Then solve for
the subsequent values of x1 , x2 , x3 , etc. , using the
following equations:
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
A. Gauss-Seidel Method
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
A. Gauss-Seidel Example
Solve the previous problem:
3x1 - 0.1x2 - 0.2x3 = 7.85
0.1x1 + 7x2 - 0.3x3 = -19.3
0.3x1 – 0.2x2 + 10x3 = 71.4
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
Convergence criterion for the Gauss-Seidel
and
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology