Kaczmarz Row Projection Trace
Solve linear equations through successive row projections and see inconsistent cycling.
Free theory articleSolve linear equations through successive row projections and see inconsistent cycling.
Concept & interpretation
The Kaczmarz method is an iterative algorithm for solving systems of linear equations. It operates by successively projecting the current guess for the solution onto the hyperplanes defined by each equation in the system. Starting from an initial guess, typically the origin, the method updates the position by calculating the orthogonal projection onto the next row's line.
Experiment
Open the Kaczmarz Row Projection Trace module to interact with this concept:
- Select the Consistent option from the System control.
- Set Steps to 4 to watch the algorithm converge geometrically to the Unique intersection.
- Switch the Order control to Reverse and set Steps to 2 to see a different path to the exact same Unique intersection at (1, 1).
- Change the System to Inconsistent and set Steps to 2 to observe the points endlessly cycle between parallel lines with a final status of Inconsistent parallel rows.