Examples
Examples
Try the starting example
- Restricted expression f(x)
- x - 1
- Bracket left
- 0
- Bracket right
- 2
- Absolute bracket-width tolerance
- 1e-08
- Maximum iterations
- 100
- Root estimate
- 1
- Function value at estimate
- 0
- Iterations used
- 1
- Remaining bracket width
- 0
- Iteration status
- Converged
- Trace: iteration, midpoint, f(midpoint), bracket width before update
- 1, 1, 0, 2
Try another scenario
- Restricted expression f(x)
- x^2 - 4
- Bracket left
- 0
- Bracket right
- 8
- Absolute bracket-width tolerance
- 1e-08
- Maximum iterations
- 100
- Root estimate
- 2
- Function value at estimate
- 0
- Iterations used
- 2
- Remaining bracket width
- 0
- Iteration status
- Converged
- Trace: iteration, midpoint, f(midpoint), bracket width before update
- 1, 4, 12, 8; 2, 2, 0, 4
Calculation method
Calculation method
Root estimate
Repeatedly bisect a bracket with opposite endpoint signs. Exact sampled zeros finish immediately.
Function value at estimate
Evaluate the same restricted function at the returned estimate.
Iterations used
Number of midpoint evaluations.
Remaining bracket width
Width after the final bracket update; zero at an exact sampled root.
Iteration status
State whether the bracket/zero stopping condition or iteration limit was reached.
Trace: iteration, midpoint, f(midpoint), bracket width before update
List each sampled midpoint and the bracket before its update.
Symbols and units
- a
- Restricted expression f(x)
- b
- Bracket left
- c
- Bracket right
- d
- Absolute bracket-width tolerance
- e
- Maximum iterations
Use the units shown. A list keeps the order in which you enter its values. The symbol ^ means “raised to the power of”.
Assumptions & limits
Assumptions & limits
- Assumes a continuous real function on the bracket; opposite endpoint signs alone do not prove continuity. Expressions permit x, pi, e, arithmetic and the documented single-argument maths functions. Inspect the residual, especially near discontinuities. Maximum 200 iterations. Restricted expressions allow x (and y for ODEs), pi, e, + - * / ^, and sin, cos, tan, sec, csc, cot, exp, log, sqrt, abs, sinh, cosh, tanh, sech, csch, coth. Trigonometric arguments are in radians. Double-precision results are approximate. Numeric tables use 12 significant digits and display components at roundoff scale (32 machine epsilons relative to their row or relevant calculation scale) as zero; scalar diagnostics retain full precision.
Are these inputs private?
These calculations run in your browser. A shared link contains the values you choose to share. Only questions submitted to the separate AI finder are sent to OpenAI.
Further reading
SciPy: bracketed root methods ↗
Background reading; this does not imply endorsement or independent review of SumFizz.