Since
the
late
the
widespread
availability
of
digital
computers
has
led
to
a
veritable
explosion
in
the
use
and
development
of
.
At
fi
rst
,
this
growth
was
somewhat
limited
by
the
cost
of
access
to
large
mainframe
computers
,
and
,
consequently
,
many
engineers
continued
to
use
simple
approaches
in
a
significant
portion
of
their
work
.
Needless
to
say
,
the
recent
evolution
of
inexpensive
personal
computers
has
given
us
ready
access
to
powerful
computational
capabilities
.
There
are
several
additional
reasons
why
you
should
study
numerical
methods
:
1
.
Numerical
methods
are
extremely
powerful
problem
-
solving
tools
.
They
are
capable
of
large
systems
of
equations
,
,
and
complicated
geometries
that
are
not
uncommon
in
engineering
practice
and
that
are
often
impossible
to
solve
analytically
.
As
such
,
they
greatly
enhance
your
problem
-
solving
skills
.
2
.
During
your
careers
,
you
may
often
have
occasion
to
use
commercially
available
prepackaged
,
or
?
canned
,
?
programs
that
involve
numerical
methods
.
The
intelligent
use
of
these
programs
is
often
predicated
on
knowledge
of
the
basic
theory
underlying
the
methods
.
3
.
Many
problems
cannot
be
approached
using
canned
programs
.
If
you
are
conversant
with
numerical
methods
and
are
adept
at
computer
programming
,
you
can
your
own
programs
to
solve
problems
without
having
to
buy
or
commission
expensive
software
.
As
summarized
,
these
are
1
.
Roots
of
Equations
.
These
problems
are
concerned
with
the
of
a
variable
or
a
parameter
that
satisfies
a
single
nonlinear
equation
.
These
problems
are
especially
valuable
in
engineering
design
contexts
where
it
is
often
impossible
to
explicitly
solve
design
equations
for
parameters
.
2
.
Systems
of
Linear
Algebraic
Equations
.
These
problems
are
similar
in
spirit
to
roots
of
equations
in
the
sense
that
they
are
concerned
with
values
that
satisfy
equations
.
However
,
in
contrast
to
satisfying
a
single
equation
,
a
set
of
values
is
sought
that
simultaneously
satisfi
es
a
set
of
algebraic
equations
.
Such
equations
arise
in
a
variety
of
problem
contexts
and
in
all
disciplines
of
engineering
.
In
particular
,
they
originate
in
the
mathematical
modeling
of
large
systems
of
elements
such
as
structures
,
electric
circuits
,
and
fluid
networks
.
However
,
they
are
also
encountered
in
other
areas
of
numerical
methods
such
as
curve
fitting
and
differential
equations
.
3
.
Optimization
.
These
problems
involve
determining
a
value
or
values
of
an
variable
that
correspond
to
a
?
best
?
or
optimal
value
of
a
function
.
Thus
,
optimization
involves
maxima
and
minima
.
Such
problems
occur
routinely
in
engineering
design
contexts
.
They
also
arise
in
a
number
of
other
numerical
methods
.
We
address
both
single
-
and
multi
-
variable
unconstrained
optimization
.
We
also
describe
constrained
with
particular
emphasis
on
linear
programming
.
4
.
Curve
Fitting
.
You
will
often
have
occasion
to
fi
t
to
data
points
.
The
techniques
developed
for
this
purpose
can
be
into
two
general
categories
:
regression
and
interpolation
.
is
employed
where
there
is
a
signifi
cant
degree
of
error
associated
with
the
data
.
Experimental
results
are
often
of
this
kind
.
For
these
situations
,
the
strategy
is
to
derive
a
single
curve
that
represents
the
general
trend
of
the
data
without
necessarily
matching
any
individual
points
.
In
contrast
,
is
used
where
the
objective
is
to
determine
intermediate
values
between
error
-
free
data
points
.
Such
is
usually
the
case
for
tabulated
information
.
For
these
situations
,
the
strategy
is
to
fit
a
curve
directly
through
the
data
points
and
use
the
curve
to
predict
the
intermediate
values
.
5
.
Integration
.
As
depicted
,
a
physical
interpretation
of
numerical
integration
is
the
determination
of
the
under
a
curve
.
Integration
has
many
applications
in
engineering
practice
.
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