UNIT 3 • NUMERICAL INTEGRATION

Romberg Integration
Trapezoidal + Richardson Extrapolation

Romberg integration starts from trapezoidal estimates with 1, 2, 4, … strips and improves them with $R(i, j) = R(i, j - 1) + \frac{R(i, j - 1) - R(i - 1, j - 1)}{4^j - 1}$ . Each column is more accurate than the one before.

THE ROMBERG TABLE2ᵏ STRIPS

\begin{gathered} R(i, 0) = T_{2^i} \ (\text{trapezoidal rule with } 2^i \text{ strips}) \\ R(i, j) = R(i, j - 1) + \frac{R(i, j - 1) - R(i - 1, j - 1)}{4^j - 1} \end{gathered}

1. TrapezoidsColumn 0: the trapezoidal rule, halving h each row (reuse the previous row's points).

2. ExtrapolateEach new column combines the entry to its left with the one above-left.

3. Read the CornerThe bottom-right entry is the answer.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Use Romberg integration with 2 levels to estimate $\int_0^1 \frac{dx}{1 + x}$ . (The exact value is $\ln 2 \approx 0.693147$ .)

STEP 01 • TRAPEZOIDAL COLUMN

R(0,0) = \tfrac{1}{2}\left[f(0) + f(1)\right] = \tfrac{1}{2}(1 + 0.5) = 0.75

R(1,0) = \tfrac{1}{2}R(0,0) + 0.5\,f(0.5) = 0.375 + 0.333333 = 0.708333

R(2,0) = \tfrac{1}{2}R(1,0) + 0.25\left[f(0.25) + f(0.75)\right] = 0.697024

STEP 02 • EXTRAPOLATE

R(1,1) = 0.708333 + \frac{0.708333 - 0.75}{3} = 0.694444

R(2,1) = 0.697024 + \frac{0.697024 - 0.708333}{3} = 0.693254

R(2,2) = 0.693254 + \frac{0.693254 - 0.694444}{15} = 0.693175

Four strips of plain trapezoids were off by 0.0039; the Romberg corner is off by only 0.000028.

FINAL CONCLUSION

∫ ≈ 0.693175 (ln 2 = 0.693147)

R(2,2) = 0.693175

The calculator below opens on this integral. Add a level to see the answer match ln 2 to six places.

LABORATORY

Interactive Romberg Calculator

LABORATORY

Romberg Engine

f(x)

Lower limit aUpper limit bLevels (1–10)

Limits can be numbers or constants such as pi or pi/2. Level k uses 2ᵏ strips.

Common Questions

How does Romberg integration work?

It works out the trapezoidal rule with 1, 2, 4, 8, … strips, then combines neighbouring estimates (Richardson extrapolation) to cancel most of their error. The bottom-right entry of the table is usually far more accurate than any single trapezoidal result.

What is Richardson extrapolation?

If two estimates have errors that shrink in a known way, a weighted combination cancels the leading error:

R(i, 1) = \frac{4R(i, 0) - R(i - 1, 0)}{3}

. That first column of corrections equals Simpson's 1/3 rule.

How many levels do I need?

For a smooth function, 3 to 5 levels usually give 6 or more correct digits. Stop when the last two diagonal entries agree to the accuracy you need.

PreviousGauss Quadrature

Unit 3•25 of 63

NextGauss Elimination

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 3 • NUMERICAL INTEGRATION

Romberg Integration
Trapezoidal + Richardson Extrapolation

THE ROMBERG TABLE2ᵏ STRIPS

\begin{gathered} R(i, 0) = T_{2^i} \ (\text{trapezoidal rule with } 2^i \text{ strips}) \\ R(i, j) = R(i, j - 1) + \frac{R(i, j - 1) - R(i - 1, j - 1)}{4^j - 1} \end{gathered}

1. TrapezoidsColumn 0: the trapezoidal rule, halving h each row (reuse the previous row's points).

2. ExtrapolateEach new column combines the entry to its left with the one above-left.

3. Read the CornerThe bottom-right entry is the answer.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Use Romberg integration with 2 levels to estimate $\int_0^1 \frac{dx}{1 + x}$ . (The exact value is $\ln 2 \approx 0.693147$ .)

STEP 01 • TRAPEZOIDAL COLUMN

R(0,0) = \tfrac{1}{2}\left[f(0) + f(1)\right] = \tfrac{1}{2}(1 + 0.5) = 0.75

R(1,0) = \tfrac{1}{2}R(0,0) + 0.5\,f(0.5) = 0.375 + 0.333333 = 0.708333

R(2,0) = \tfrac{1}{2}R(1,0) + 0.25\left[f(0.25) + f(0.75)\right] = 0.697024

STEP 02 • EXTRAPOLATE

R(1,1) = 0.708333 + \frac{0.708333 - 0.75}{3} = 0.694444

R(2,1) = 0.697024 + \frac{0.697024 - 0.708333}{3} = 0.693254

R(2,2) = 0.693254 + \frac{0.693254 - 0.694444}{15} = 0.693175

Four strips of plain trapezoids were off by 0.0039; the Romberg corner is off by only 0.000028.

FINAL CONCLUSION

∫ ≈ 0.693175 (ln 2 = 0.693147)

R(2,2) = 0.693175

The calculator below opens on this integral. Add a level to see the answer match ln 2 to six places.

LABORATORY

Interactive Romberg Calculator

LABORATORY

Romberg Engine

f(x)

Lower limit aUpper limit bLevels (1–10)

Limits can be numbers or constants such as pi or pi/2. Level k uses 2ᵏ strips.

Common Questions

How does Romberg integration work?

What is Richardson extrapolation?

If two estimates have errors that shrink in a known way, a weighted combination cancels the leading error:

R(i, 1) = \frac{4R(i, 0) - R(i - 1, 0)}{3}

. That first column of corrections equals Simpson's 1/3 rule.

How many levels do I need?

For a smooth function, 3 to 5 levels usually give 6 or more correct digits. Stop when the last two diagonal entries agree to the accuracy you need.

PreviousGauss Quadrature

Unit 3•25 of 63

NextGauss Elimination

Romberg Integration Trapezoidal + Richardson Extrapolation

Step-by-Step Worked Example

∫ ≈ 0.693175 (ln 2 = 0.693147)

Interactive Romberg Calculator

Romberg Engine

Common Questions

NETZ Engine

Romberg Integration Trapezoidal + Richardson Extrapolation

Step-by-Step Worked Example

∫ ≈ 0.693175 (ln 2 = 0.693147)

Interactive Romberg Calculator

Romberg Engine

Common Questions

Romberg Integration
Trapezoidal + Richardson Extrapolation

Romberg Integration
Trapezoidal + Richardson Extrapolation