What are numerical methods, and why are they used instead of analytical solutions?
Numerical methods are algorithms that use approximations and iterative procedures to solve mathematical problems. They are used when analytical (closed-form) solutions do not exist, are too difficult to obtain, or are impractical for real engineering problems.
Fill in the blank: Numerical methods are an __________ tool, not a __________ abstraction.
engineering, mathematical
Numerical methods always give the exact solution to a problem. (T/F)
False, give approximate solutions with controllable error.
Give two examples of engineering problems that require numerical methods.
Flow depth in open channel (root finding)
Heat flow in concrete curing (finite differences)
Beam deflection (numerical integration)
Truss/frame analysis (systems of equations)
What is the difference between true error and approximate error?
True error: difference between the exact value and the numerical value
Approximate error: difference between successive numerical estimates (used when true value is unknown)
T/F: Reducing step size or tolerance will always reduce error without consequences
False, can increase computation time and round-off error
Which type of error is caused by limited precision in computers?
A) Truncation error
B) Round-off error
C) Modeling error
D) Iteration error
B) Round-off error
What does it mean to bracket a root?
To choose two points xl and xu such that f(xl)*f(xu)<0 which guarantees a root exists between them.
T/F: Bisection method is guaranteed to find a root if the initial interval brackets a root.
True
Why does the bisection method always converge (if properly bracketed)?
Because each iteration halves the interval containing the root, and the Intermediate Value Theorem guarantees the root remains inside the interval.
Give one advantage and one disadvantage of bisection.
Advantage: Guaranteed convergence
Disadvantage: Slow (linear convergence)
False position differs from bisection because it:
A) Does not require bracketing
B) Uses derivatives
C) Uses linear interpolation instead of midpoint
D) Always converges faster
C)
What is a common problem with the false position method?
One endpoint may remain fixed for many iterations, slowing convergence.
T/F: Open methods require a bracketing interval
False
Why are open methods usually faster than bracketing methods?
They use local slope or function behavior to jump toward the root rather than shrinking an interval.
Why is fixed-point iteration considered “simple but dangerous”?
Because convergence is not guaranteed and depends heavily on the choice of 𝑔(𝑥).
A fixed point satisfies the equation __________.
g(x)=x
What condition ensures convergence of fixed-point iteration?
∣g′(x)∣<1near the root
Give two reasons Newton’s method is powerful and two reasons it can fail.
Pros:
Very fast (quadratic convergence)
Efficient near the root
Cons:
Requires derivative
Can diverge or jump to wrong root
Fails if 𝑓′(𝑥)=0
T/F: Newton’s method is always better than bisection.
False
What does it mean for a system to be ill-conditioned?
Small changes in inputs cause large changes in the solution.
Which operation does NOT change the solution of a linear system?
A) Swapping rows
B) Multiplying a row by a constant
C) Adding a multiple of one row to another
D) Rounding coefficients
D)
Why is partial pivoting used in Gaussian elimination?
To reduce round-off error by avoiding division by small pivot elements.
T/F: Row scaling improves numerical stability but changes the system.
False — it improves stability without changing the solution.
