Solve steady distributions from transition matrices using flexible controls. Review errors, iterations, and residual quality. Model long term probabilities clearly for every named state.
Enter a row-stochastic transition matrix. Each row must sum to 1. You may separate values with spaces, commas, or semicolons.
| State | Transition Row | Example Stationary Probability |
|---|---|---|
| State A | 0.70, 0.20, 0.10 | 0.478723 |
| State B | 0.30, 0.40, 0.30 | 0.265957 |
| State C | 0.25, 0.25, 0.50 | 0.255319 |
This sample chain is irreducible and includes self-loops, so it is a practical example for long-run probability analysis.
Stationary distribution condition:
πP = π
Normalization constraint:
Σ πᵢ = 1
Power iteration update:
xk+1 = xkP
Residual check:
Residual = ‖πP − π‖₁
The calculator solves the steady-state vector for a finite Markov chain. Linear solve handles the balance equations directly. Power iteration repeatedly propagates an initial distribution through the transition matrix until the change falls below the chosen tolerance.
It represents long-run state probabilities for a Markov chain. If the chain mixes well, repeated transitions drive many starting distributions toward this stable probability vector.
Each row lists all possible next-state probabilities from one current state. Since one of those outcomes must occur, the full row must total exactly 1.
Use power iteration when you want to inspect convergence behavior, test sensitivity to an initial distribution, or work with larger matrices where iterative methods are convenient.
Linear solve is often faster for small and medium matrices. It directly solves the stationary equations and usually provides a clean reference solution for comparison.
Yes. Reducible chains can admit multiple stationary distributions. In those cases, convergence and uniqueness depend on the chain’s communicating classes and the initial distribution.
The residual measures how closely the reported vector satisfies πP = π. Smaller residuals indicate a more accurate stationary solution.
Slow mixing, near-reducibility, periodic behavior, or a very strict tolerance can require many iterations. Increasing the iteration cap often helps reveal the trend.
It provides helpful diagnostics, including a strong connectivity test and a self-loop based aperiodicity check. Those are practical indicators, not a full symbolic proof.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.