SME
Numerical Methods for Engineers · Problem type
One step of the power method
Use this for “Start the power method on A = [6 2; 2 3] from x₀ = (1, 1). Find Ax₀ and the Rayleigh estimate of the largest eigenvalue.”
Diagram
A is 2 × 2 and x₀ is 2 × 1, so Ax₀ is 2 × 1.
Row 1 of A times column 1 of x₀ gives entry (1, 1): c₁₁ = 6 × 1 + 2 × 1 = 8.
Tap an entry of Ax₀ to see its row and column.
This is an example. Type your own values to start a new problem.
=
λ
Rayleigh estimate
calculated
Formulas
y₁
=
a₁₁
×
x₁
+
a₁₂
×
x₂
8 = 6 × 1 + 2 × 1
y₂
=
a₂₁
×
x₁
+
a₂₂
×
x₂
5 = 2 × 1 + 3 × 1
λ
=
x₁
×
y₁
+
x₂
×
y₂
x₁
2
+
x₂
2
6.5
=
1 × 8 + 1 × 5
1
2
+
1
2
Assumptions
•
One eigenvalue is largest in size, and x₀ holds some of its eigenvector.
•
Repeat with x = y (scaled) until the estimate λ settles; this page works one step.
•
The Rayleigh quotient λ = (x · Ax) ÷ (x · x) needs x₀ not all zeros.
Step-by-step
WE KNOW
a₁₁ = 6, a₁₂ = 2, a₂₁ = 2, a₂₂ = 3, x₁ = 1, x₂ = 1
FIND
product, first entry (y₁), product, second entry (y₂), Rayleigh estimate (λ)
Step 1 · Find product, first entry (y₁)
Use y₁ = a₁₁ × x₁ + a₁₂ × x₂
Row 1 of A times the column x₀: multiply entry by entry and add.
y₁
=
6
×
1
+
2
×
1
y₁
=
8
Step 2 · Find product, second entry (y₂)
Use y₂ = a₂₁ × x₁ + a₂₂ × x₂
Row 2 of A times the column x₀: multiply entry by entry and add.
y₂
=
2
×
1
+
3
×
1
y₂
=
5
Step 3 · Find Rayleigh estimate (λ)
Use λ = (x₁ × y₁ + x₂ × y₂) ÷ (x₁² + x₂²)
The Rayleigh quotient: x₀ dotted with Ax₀, over x₀ dotted with itself.
λ
=
1 × 8 + 1 × 5
1
2
+
1
2
λ
=
13
1 + 1
λ
=
13
2
λ
=
6.5
Check
Put the numbers back in. Both sides should match.
8 = 6 × 1 + 2 × 1 ✓
5 = 2 × 1 + 3 × 1 ✓
6.5
=
1 × 8 + 1 × 5
1
2
+
1
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.