Matrix Chain Multiplication Calculator Online

Enter dimensions and compare candidate chain multiplication costs. View splits, tables, and export results easily. Choose parenthesizations that reduce computation across larger matrix workloads.

Calculator Form

Enter comma-separated dimensions p0, p1, p2, ... , pn.
A chain using prefix A becomes A1, A2, A3, and so on.
This text appears beside the final minimum cost.

Example Data Table

Example Dimension Vector Matrix Chain Expected Minimum Cost
Classic DP Example 30, 35, 15, 5, 10, 20, 25 A1: 30×35, A2: 35×15, A3: 15×5, A4: 5×10, A5: 10×20, A6: 20×25 15,125
Short Chain 10, 20, 30, 40 A1: 10×20, A2: 20×30, A3: 30×40 18,000
Balanced Sizes 5, 10, 3, 12, 5, 50, 6 A1: 5×10, A2: 10×3, A3: 3×12, A4: 12×5, A5: 5×50, A6: 50×6 2,010

Formula Used

Matrix chain multiplication uses dynamic programming. It does not change the matrix order. It only changes the parenthesization.

Base case: m[i,j] = 0 when i = j

Recurrence: m[i,j] = min(m[i,k] + m[k+1,j] + p[i-1] × p[k] × p[j]) for every k from i to j-1

Here, p is the dimension vector. If the vector is 30, 35, 15, 5, then the chain is 30×35, 35×15, and 15×5.

The merge term p[i-1] × p[k] × p[j] counts the scalar operations needed after the left and right subchains are already solved.

The split table stores the best k for each subchain. That table helps rebuild the final optimal parenthesization.

How to Use This Calculator

  1. Enter the matrix dimensions as comma-separated values.
  2. Make sure each adjacent pair defines a valid matrix chain.
  3. Choose a matrix label prefix if you want custom names.
  4. Set a custom cost unit label if needed.
  5. Enable candidate split review for deeper analysis.
  6. Press the calculate button.
  7. Read the result block above the form.
  8. Review the optimal order, DP tables, and graph.
  9. Download the result as CSV or PDF if required.

FAQs

1) What does matrix chain multiplication optimization do?

It finds the multiplication order that minimizes scalar operations for compatible matrices. The matrix sequence stays fixed. Only the grouping changes.

2) Why can two valid orders have different costs?

Matrix multiplication is associative, but intermediate matrix sizes can change with grouping. Different temporary sizes cause different scalar multiplication counts.

3) Does this calculator multiply actual matrix values?

No. It optimizes dimension-based multiplication order only. It focuses on cost, split points, dynamic tables, and the best parenthesization.

4) What input format should I enter?

Enter positive integers separated by commas or spaces. Example: 30,35,15,5,10,20,25. That creates six compatible matrices.

5) Why do I enter one more dimension than matrices?

A chain of n matrices needs n+1 dimensions. Each matrix uses one starting dimension and one ending dimension from the vector.

6) What does the best split value mean?

It is the index k that divides one subchain into two smaller parts. The recurrence tests all valid splits and keeps the cheapest.

7) What does the graph show?

The graph plots minimum solved costs for subchains by length. It helps you see how complexity grows as the chain becomes longer.

8) Can I save the result for reports?

Yes. Use the CSV or PDF buttons after calculation. The exported file includes the summary, matrix list, and optimization steps.

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