Tip: try the example values below, then press Solve.
Example data table
Try this 3×3 system (paste each value into the matching fields):
| Row | a11 | a12 | a13 | b1 |
|---|---|---|---|---|
| 1 | 2+i | -1 | 3-2i | 4+i |
| 2 | 1 | -2+i | 1+i | -1+3i |
| 3 | 0.5 | 1- i | -1 | 2 |
Formula used
This calculator solves a complex-valued linear system:
A x = b
- Method: Gaussian elimination with partial pivoting (largest |pivot| in each column).
- Back substitution: solves the resulting upper-triangular system.
- Determinant: product of pivots, adjusted by row swaps.
- Residual: r = Ax − b, reported with ∞-norm max |ri|.
- Condition estimate: cond∞(A) ≈ ||A||∞ · ||A⁻¹||∞ when inversion succeeds.
Complex arithmetic uses (a+bi)(c+di) and division by c²+d².
How to use this calculator
- Select the system size n (2 to 6).
- Enter each coefficient aij and constant bi.
- Use formats like 1+2i, -3i, or 4.
- Optionally set ε to control pivot detection sensitivity.
- Click Solve System to view results above the form.
- Download CSV or PDF after a successful solve.
FAQs
1) What does “complex system” mean here?
It means the coefficients and constants can include an imaginary part, like 3−2i. The solver handles complex arithmetic and returns complex solutions.
2) What input formats are accepted?
Use a, bi, a+bi, or a−bi. Examples: 2, -1.5, 4i, -i, 3+0.2i. Spaces are ignored.
3) Why do I see a “singular or ill-conditioned” message?
If a pivot magnitude falls below ε, the matrix may be singular or too close to singular. Try increasing ε slightly or check that your equations are independent.
4) What is the residual and why does it matter?
The residual r = Ax − b measures how well the computed solution satisfies the equations. Smaller residuals generally indicate better numerical accuracy.
5) What does the condition estimate tell me?
A larger condition number suggests the solution may be sensitive to small input changes. It is an estimate, not a guarantee, especially near singularity.
6) How many equations can I solve?
You can solve systems from 2×2 up to 6×6. Larger systems are possible, but web forms become bulky and numerical errors can grow.
7) Can this solve nonlinear complex systems?
This tool solves linear systems only. For nonlinear systems, you would typically use iterative methods like Newton’s method with complex derivatives.
8) Why does the PDF look simple?
The PDF export is intentionally lightweight for portability. It prints key results as text, making it easy to save, share, and archive.