Calculator inputs
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Formula used
QR avoids directly forming the normal equations. Orthogonal transformations usually improve numerical stability. Pivoting helps reveal dependent matrix columns during computation.
How to use
- Select a least-squares or decomposition mode.
- Choose a QR decomposition method.
- Set matrix dimensions and build the editor.
- Enter matrix A and right-hand side B.
- Set precision, tolerance, and display preferences.
- Press calculate and inspect every reported result.
- Copy, print, or download the completed calculation.
Example data
| x | 1 | b | Purpose |
|---|---|---|---|
| 1 | 1 | 6 | First observation |
| 2 | 1 | 5 | Second observation |
| 3 | 1 | 7 | Third observation |
| 4 | 1 | 10 | Fourth observation |
Frequently asked questions
What problem does this calculator solve?
It solves overdetermined linear systems using QR factorisation. The result minimises the squared residual error. It also reports diagnostics supporting careful result review.
Which QR method should I choose?
Householder reflections provide a strong general default. Modified Gram–Schmidt supports clear educational steps. Pivoting helps when matrix columns are dependent.
Why avoid normal equations?
Normal equations square the matrix condition number. That can increase floating-point error noticeably. QR methods usually provide more reliable numerical solutions.
What does rank deficiency mean?
Rank deficiency indicates dependent or nearly dependent columns. A unique coefficient vector may not exist. Pivoted QR identifies influential and dependent variable columns.
Can I solve weighted least squares?
Yes, select weighted least squares mode. Enter one positive weight per observation. The calculator scales rows before QR decomposition internally.
Does it support several response vectors?
Yes, increase the right-hand-side count before building. Each response receives its own coefficient vector. Shared QR factors reduce repeated numerical work substantially.
How is numerical rank detected?
Diagonal values of R are compared against tolerance. Small values indicate missing numerical independence. Adjust tolerance when data scales differ significantly.
What do reconstruction errors show?
They compare QR against the entered matrix. Small errors support a correct factorisation. Larger errors may indicate scaling or precision problems.
Can results be exported?
Yes, copy complete text or download files. CSV, LaTeX, PDF, and printing are supported. Exports include matrices, metrics, and verification details.