In one line
Regula Falsi joins the points \((a,f(a))\) and \((b,f(b))\) on opposite sides of the axis by a straight chord and takes where the chord cuts the \(x\)-axis as the next approximation, always keeping the root bracketed.
Explained simply
Suppose at 9 am the temperature is \(-2^\circ\)C and at 11 am it is \(+6^\circ\)C. When did it cross zero? A straight-line guess says 9:30 am, a quarter of the way. That is false position: pretend the curve is a straight line between two points, find where it hits zero, then repeat with the smaller interval. It is called "false" because the curve is not really straight, but the guesses improve.
Technically, if \(f(a)f(b)<0\), the chord meets the axis at \(x_1=\dfrac{af(b)-bf(a)}{f(b)-f(a)}\). If \(f(x_1)\) has the sign of \(f(a)\), replace \(a\) by \(x_1\); otherwise replace \(b\). Unlike Newton–Raphson, it needs no derivative and never loses the root, but it usually converges more slowly (linearly).
How it works
- Find \(a,b\) with \(f(a)<0<f(b)\) (or the other way).
- Compute \(x_1=\dfrac{af(b)-bf(a)}{f(b)-f(a)}\) and \(f(x_1)\).
- Keep the sub-interval where the sign changes.
- Repeat until successive values agree to the required accuracy.
Key formula
\(x_{\text{new}}=\dfrac{af(b)-bf(a)}{f(b)-f(a)}\), where \(f(a)\) and \(f(b)\) have opposite signs.
Quick example
\(f(x)=x^3-2x-5\), \(a=2\) (\(f=-1\)), \(b=3\) (\(f=16\)).
| Step | \(a\) | \(b\) | \(x\) | \(f(x)\) |
|---|---|---|---|---|
| 1 | 2 | 3 | \(\frac{35}{17}=2.05882\) | \(-0.3908\) |
| 2 | 2.05882 | 3 | 2.08126 | \(-0.1472\) |
| 3 | 2.08126 | 3 | 2.08964 | \(-0.0547\) |
| 4 | 2.08964 | 3 | 2.09274 | \(-0.0202\) |
The values creep towards 2.0946. Notice the end \(b=3\) stays fixed, which is why Regula Falsi is slower than Newton–Raphson here (which needed 3 steps).
Common mistakes
- Using the formula with \(f(a)\) and \(f(b)\) of the same sign.
- Replacing the wrong end of the interval after computing \(f(x)\).
- Mixing up the formula as \(\dfrac{af(a)-bf(b)}{\dots}\).
How MU asks it
Common 5/6-mark question (also a SCILAB tutorial): "Find a root of \(x^3-4x-9=0\) by Regula Falsi method correct to three decimal places" or "Solve \(xe^x=\cos x\) using the method of false position". Present a table with \(a\), \(b\), \(f(a)\), \(f(b)\), \(x\), \(f(x)\) for each step and state the final root.