Calculator Inputs
Formula Used
1) Total nonzeros
nnz = N × average_nonzeros_per_row
2) Matrix density
density(%) = (nnz / N²) × 100
3) CSR storage
CSR bytes = nnz × (value_bytes + index_bytes) + (N + 1) × index_bytes
4) Workspace memory
workspace = vector_count × N × value_bytes
5) Total memory
total_memory = (CSR bytes + workspace) × (1 + overhead_percent / 100)
6) Effective iterations
effective_iterations = baseline_iterations × (1 - preconditioner_gain / 100)
7) Flops per iteration
CG ≈ 2 × nnz + 12 × N
BiCGSTAB ≈ 4 × nnz + 24 × N
GMRES(m) ≈ 2 × nnz + (4m + 10) × N
8) Total work and runtime
total_flops = flops_per_iteration × effective_iterations
solve_time = total_flops / achieved_GFLOPS / 10⁹
9) Residual decay estimate
residual(k) = initial_residual × decay_ratio^k
How to Use This Calculator
- Choose the physics model closest to your simulation or discretized PDE system.
- Select a solver type. Use CG for symmetric positive definite systems, BiCGSTAB for general nonsymmetric systems, and GMRES when stronger robustness matters.
- Enter the number of unknowns and the average nonzeros per row from your sparse stencil, mesh, or assembly pattern.
- Provide baseline iterations and the expected preconditioner gain. Higher gain lowers the effective iteration count.
- Enter tolerance, initial residual, and RHS norm to estimate the convergence path and relative residual.
- Set achieved throughput, precision bytes, index bytes, and memory overhead to reflect your hardware and implementation choices.
- Click Calculate Solver Metrics to display results above the form, review the table, inspect the Plotly graph, and export CSV or PDF.
- Compare scenarios by changing solver type, restart length, or precision and recalculating.
Example Data Table
| Parameter | Example Value | Meaning |
|---|---|---|
| Physics Model | Heat Diffusion | Typical sparse system from a discretized diffusion equation. |
| Solver Type | CG | Good fit for symmetric positive definite matrices. |
| Unknowns (N) | 50,000 | Total equations or degrees of freedom. |
| Average Nonzeros per Row | 7 | Typical five-point or seven-point style sparsity pattern. |
| Baseline Iterations | 180 | Expected iterations without stronger tuning. |
| Preconditioner Gain | 30% | Approximate iteration reduction from preconditioning. |
| Target Tolerance | 1.0e-8 | Requested stopping threshold. |
| Achieved Throughput | 24 GFLOP/s | Observed or assumed machine performance. |
| Precision Bytes | 8 | Double precision real values. |
| Index Bytes | 4 | Standard 32-bit row and column indexing. |
Frequently Asked Questions
1) What does this sparse matrix solver calculator estimate?
It estimates nonzeros, density, CSR storage, workspace memory, floating-point work, solve time, and residual decay for common iterative sparse solvers used in physics simulations.
2) Why is this useful for physics problems?
Physics models from Poisson, diffusion, wave, and finite element systems often create very large sparse matrices. Planning memory and solver effort early helps avoid failed jobs and poor runtime choices.
3) Does it run an actual Fortran solver?
No. It provides engineering estimates that help size and compare solver options. You can use the results to guide real Fortran implementations, benchmarks, and production job planning.
4) When should I choose CG?
Choose CG for symmetric positive definite systems, such as many diffusion or Poisson discretizations. It usually offers lower memory use and less work per iteration than broader nonsymmetric methods.
5) When is BiCGSTAB or GMRES better?
Use BiCGSTAB or GMRES when your matrix is nonsymmetric, indefinite, or harder to converge. GMRES is more robust, while BiCGSTAB often uses less memory and can be faster.
6) What does average nonzeros per row represent?
It approximates the stencil width or element connectivity in your sparse system. Larger values increase storage, matrix-vector cost, and total solver work.
7) Why does precision size matter?
Precision controls memory footprint and sometimes performance. Double precision usually improves numerical reliability, while single precision can reduce memory and bandwidth demands.
8) Are the runtime and residual values exact?
No. They are informed estimates. Real performance depends on matrix ordering, cache behavior, preconditioner quality, hardware bandwidth, compiler settings, and solver implementation details.