Calculator Inputs
Example Data Table
This sample uses 2×2 blocks for a 4×4 by 4×4 multiplication.
| Field | Sample Value | Notes |
|---|---|---|
| Dimensions | A: 4×4, B: 4×4 | Inner dimension 4 matches. |
| Block Sizes | Rows: 2, Inner: 2, Cols: 2 | Creates eight block updates. |
| Matrix A First Row | 1 2 0 1 | Enter one row per line. |
| Matrix B First Row | 2 1 3 0 | Spaces or commas accepted. |
| Expected C(1,1) | 4 | Computed as row1(A) · col1(B). |
Formula Used
Standard matrix multiplication computes each result cell using: Ci,j = Σ Ai,k × Bk,j for k = 1...n.
Block multiplication applies the same rule in chunks, which is useful for performance and memory locality: CI,J += AI,K × BK,J. Here I, J, K represent row, column, and inner index ranges defined by your block sizes.
Scalar operation estimates shown by the tool use: m × n × p multiplications and m × p × (n − 1) additions.
How to Use This Calculator
- Enter matrix dimensions: rows and columns for Matrix A, then columns for Matrix B.
- Set block sizes for rows, inner dimension, and columns. They can be smaller than dimensions.
- Paste Matrix A and Matrix B values. Use one row per line.
- Separate values with spaces, commas, or semicolons.
- Choose decimal precision and press Submit.
- Review the result matrix above the form, plus the block execution log.
- Use CSV or PDF buttons to export the result for reports or coursework.
Block Processing and Performance Context
Block matrix multiplication improves runtime behavior by processing tiles instead of entire rows and columns continuously. The mathematics remains identical, but memory access becomes more predictable, which matters for larger matrices. This calculator demonstrates that structure clearly by listing each block update. Users can see how A and B partitions contribute to a specific C partition, making the algorithm easier to teach, debug, and validate in practical numerical workflows and classroom demonstrations.
Dimension Rules and Edge Blocks
Matrix compatibility still follows the standard rule: if A is m by n, B must be n by p, and the output becomes m by p. Block dimensions only define grouping, not correctness. The tool accepts uneven dimensions and automatically handles edge tiles when dimensions are not exact multiples of block sizes. That makes it useful for realistic datasets, where matrices often arrive in irregular shapes without convenient padding or preprocessing safely.
Reading the Summary Metrics
The operation summary supports workload estimation and review. Scalar multiplications are reported as m times n times p, while scalar additions are m times p times n minus one. These counts remain constant for the same matrices, regardless of tile settings. In contrast, the number of block steps changes with block rows, block inner, and block columns. This helps users compare granularity, logging detail, and expected loop overhead quickly across configurations efficiently.
Selecting Practical Block Sizes
Choosing block sizes depends on matrix shape and practical objectives. For learning, small symmetric blocks make the execution log easier to read. For experimentation, users can try wider column blocks or smaller inner blocks to observe how ranges shift in the log. Although this calculator does not benchmark hardware speed directly, it gives a reliable way to verify that different partition strategies produce the same numerical result under consistent precision settings repeatably.
Exports and Documentation Workflow
The calculator also supports reporting and audit tasks. CSV export is helpful for spreadsheet checks, regression comparisons, and bulk documentation. PDF export captures the result matrix and block execution log in a shareable format for reviews or assignments. The example table reduces input mistakes, while the formula and usage sections connect the tile process to the standard summation definition. Together, these features provide a professional workflow for matrix computation, explanation, and documentation.
FAQs
1. What does block matrix multiplication change?
It changes execution order only. The calculator groups operations into tiles, but the final matrix equals standard multiplication when dimensions are compatible and values are identical.
2. Can block sizes be larger than dimensions?
Yes. The calculator clips each block range at matrix boundaries, so oversized block settings still work. They simply behave like full-range loops for that dimension.
3. Why is my multiplication rejected?
Matrix A columns must equal Matrix B rows. If those inner dimensions differ, multiplication is undefined, and the calculator correctly shows a compatibility error.
4. Do block sizes affect the final answer?
No. Block sizes affect grouping, log detail, and practical performance behavior, not the mathematical result. Different valid partitions should produce the same matrix C.
5. What format should I use for matrix input?
Enter one matrix row per line. Separate numbers with spaces, commas, or semicolons. The calculator validates row counts and column counts before running.
6. When should I export CSV versus PDF?
Use CSV for spreadsheet analysis, comparison testing, and data pipelines. Use PDF for reports, assignments, reviews, or sharing a readable record of results and block steps.