WorksheetsFixed-Point Iteration (CE Applications)
Total questions: 50
Worksheet time: 25mins
In fixed-point iteration, convergence is guaranteed if:
|g'(x)|>1 near the root
|g'(x)|=1 at the root
|g'(x)|<1 near the root
g(x) is continuous only
The equation f(x)=0 is rewritten as x=g(x). Which choice most strongly affects convergence?
Initial guess only
Algebraic form of g(x)
Degree of polynomial
Number of iterations
Fixed-point iteration may fail even if the root exists because:
The function is nonlinear
The derivative of g(x) exceeds unity
The function is continuous
The root is unique
For convergence at x=r, the sufficient condition is:
g(r)=0
f'(r)=0
|g'(r)|<1
g'(r)=1
Compared with Newton–Raphson, fixed-point iteration is generally:
Faster and more accurate
Slower but more stable
Faster without derivatives
Always divergent
Fixed-point iteration is classified as a:
Bracketing method
Direct method
Open method
Matrix method
Which transformation is MOST likely to diverge?
x = √(5 − x)
x=(5−x2)(1/3)
x = 5 − x²
x = 5/(x + 1)
Linear convergence implies that the error:
Squares each iteration
Reduces by a constant factor
Vanishes instantly
Oscillates indefinitely
If g'(r) = -0.6, the iteration will:
Diverge
Oscillate with decreasing amplitude
Oscillate with increasing amplitude
Converge monotonically
A poor choice of iteration function may lead to:
Faster convergence
Guaranteed convergence
Divergence or oscillation
Root duplication
A hydraulic equation reduces to x=(10+x)(1/2) Using x₀ = 3, compute x₁.
3.32
3.61
3.74
4.00
Given x = 1/2 (7 - 2/x), with x₀ = 2, find x₁.
2.00
2.50
3.00
2.75
Settlement analysis leads to x = √(12 - x) Which initial guess ensures convergence?
x₀ = 10
x₀ = 5
x₀ = 1
Any value
For the flow equation x = ln(6 + x), evaluate x₁ if x₀ = 1.
1.61
1.79
1.95
2.08
A beam deflection equation yields x = 20 / (x + 4) If x₀ = 2, find x₂.
2.50
3.00
3.33
2.86
For g(x)=cosx, which interval ensures convergence?
[0,3]
[0,1]
[1,3]
Any real number
A Manning equation is rearranged as x = (0.02x3+1)(1/4) The convergence is MOST sensitive to:
Channel slope
Initial guess
Roughness coefficient
Number of iterations
Given x = 4/(1+x), compute x3 for x0 = 1.
1.60
1.33
1.50
1.43
Fixed-point iteration fails for x2−4 because:
No real root exists
|g'(x)| > 1
Function is discontinuous
Root is complex
A geotechnical equation gives x = ln(10x) Convergence requires:
x > 0
x < 0
x = 1
Any real x
If |g'(x)|=0.95, the method will converge:
Slowly
Rapidly
Quadratically
Not at all
In seepage analysis, the equation is x=(15−2x)1/2 . What is the risk?
Guaranteed divergence
Oscillatory convergence
No real root
Complex iteration
For x= e−x , which is TRUE?
Always diverges
Converges for small x₀
Requires bracketing
Has no root
Structural analysis yields x=x210 Why is this unstable?
A. Division by zero
B. High curvature near root
C. |g'(x)|>1
D. Root multiplicity
Which transformation improves convergence for x3−5x+1=0 ?
x=x2−51
(5x−1)1/3
x=5x3+1
x = sqrt(5x - 1)
If iteration oscillates with increasing amplitude, then:
g'(x) > 1
g'(x) < -1
g'(x) = 0
g'(x) = 1
A drainage problem leads to x = (6 + x)/4 What type of convergence occurs?
Divergence
Linear convergence
Quadratic convergence
Oscillation
Fixed-point iteration stops when:
Root is exact
Relative error is small
xₙ₊₁=0
g(x)=0
If g′(r)=0, convergence is:
Linear
Sublinear
Quadratic
Impossible
A foundation settlement equation yields slow convergence. Best remedy?
Increase iterations
Change tolerance
Reformulate g(x)
Use same initial guess
Which method is MOST reliable if fixed-point diverges?
Secant
Bisection
Newton–Raphson
Fixed-slope
Fixed-point iteration is NOT ideal when:
Derivatives are unavailable
Function is smooth
For x=ln(5/x), the major limitation is:
Logarithmic instability
Domain restriction
Infinite roots
Zero derivative
A soil bearing equation converges slowly because:
Multiple roots
Flat slope near root
Discontinuity
Wrong tolerance
Fixed-point iteration is equivalent to Newton–Raphson when:
g'(x)=1
Linear functions only
Tangent slope is constant
Iteration function is linearized
What ensures monotonic convergence?
0 < g'(x) < 1
g'(x) < -1
g'(x) = 0
g'(x) > 1
If x_{n+1} = g(x_n) overshoots alternately, then:
g'(x) > 0
g'(x) < 0
g'(x) = 0
g'(x) = 1
A beam equation requires stable convergence. Best practice is to:
Use trial-and-error
Use graphical method
Check derivative condition
Increase iterations
Which CE problem is LEAST suitable for fixed-point iteration?
Flow depth
Soil settlement
Linear truss analysis
Pipe friction factor
If convergence is conditional, it means:
Always convergent
Convergent only for some x_0
No real solution
Quadratic order
A uniform flow analysis leads to the equation: x=(12−0.5x)(1/2) Using fixed-point iteration with x₀ = 3.0, determine x₂.
2.82
2.90
2.96
3.04
The settlement equation is transformed to: x = ln(8 + 2x). Using x₀ = 1.5, compute x₁.
2.08
2.12
2.18
2.23
The Colebrook-type equation is simplified to: x = 1 / sqrt(2.5 + ln(x + 1)). Using x₀ = 0.40, determine x₁.
0.48
0.51
0.54
0.58
A nonlinear deflection equation is rearranged as: x = 15 / (x + 5) Using x₀ = 2.0, compute x₃.
2.11
2.19
2.27
2.33
The iteration function is: x = √(20 - 3x) Which condition MOST threatens convergence?
Negative root
Large initial guess
Oscillatory behavior due to g'(x)<0
Multiple roots
The head equation becomes: x = ln(10x). Using x₀ = 1.8, compute x₁.
A. 2.79
B. 2.83
C. 2.89
D. 2.94
For a fixed-point function with g'(r) = -0.92 The convergence behavior is best described as:
Divergent
Monotonic convergence
Oscillatory convergence with decreasing error
Oscillatory divergence
The equation governing flow net spacing is: x = (5 + x^2)/6 Evaluate x2 using x0 = 1.0.
1.17
1.20
1.25
1.29
A fixed-point iteration converges extremely slowly in a settlement problem. Which modification MOST improves convergence?
Increasing iteration count
Tightening tolerance
For a nonlinear CE equation where |g'(x)|≈1 near the root, the MOST appropriate numerical approach is to:
Continue fixed-point iteration
Switch to bisection method
Increase initial guess magnitude
Reduce iteration step size
