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IMPLEMENTATION_SUMMARY_CGS.md

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CGS (Conjugate Gradient Squared) Re-enablement Summary

Objective

Re-enable the CGS (Conjugate Gradient Squared) iterative linear system solver that was commented out in PR #88 during -Wall -Werror compliance work.

Changes Implemented

1. Source Code Changes (src/Numeric/LinearAlgebra/Sparse.hs)

Uncommented Functions:

  • cgsInit (lines 917-920): Initializes the CGS solver state

    cgsInit :: LinearVectorSpace (SpVector a) =>
     MatrixType (SpVector a) -> SpVector a -> SpVector a -> CGS a
cgsInit aa b x0 = CGS x0 r0 r0 r0 where
  r0 = b ^-^ (aa #> x0)    -- residual of initial guess solution

  • cgsStep (lines 922-933): Performs one CGS iteration step

    cgsStep :: (V (SpVector a), Fractional (Scalar (SpVector a))) =>
     MatrixType (SpVector a) -> SpVector a -> CGS a -> CGS a
cgsStep aa rhat (CGS x r p u) = CGS xj1 rj1 pj1 uj1
    where
    aap = aa #> p
    alphaj = (r `dot` rhat) / (aap `dot` rhat)
    q = u ^-^ (alphaj .* aap)
    xj1 = x ^+^ (alphaj .* (u ^+^ q))         -- updated solution
    rj1 = r ^-^ (alphaj .* (aa #> (u ^+^ q))) -- updated residual
    betaj = (rj1 `dot` rhat) / (r `dot` rhat)
    uj1 = rj1 ^+^ (betaj .* q)
    pj1 = uj1 ^+^ (betaj .* (q ^+^ (betaj .* p)))

  • Show instance (lines 936-940): Display CGS state for debugging

Module Exports:

Added to the export list:

  • CGS(..) - The CGS data type with all fields
  • cgsInit - Initialization function
  • cgsStep - Iteration function

2. Test Suite Changes (test/LibSpec.hs)

New Test Specification: specCGS

Comprehensive test suite with multiple levels of verification:

Unit Tests:

  1. Initial State Test: Verifies cgsInit creates correct initial state

    • Checks residual r₀ = b - Ax₀
    • Verifies p₀ = u₀ = r₀

  2. Iteration Test: Verifies cgsStep executes without errors

    • Confirms state is updated
    • Validates dimensions are preserved

Integration Tests: 3. 2x2 Dense System: Tests convergence on small dense system

  • Matrix: aa0 (2×2)
  • Known solution: x0true
  • Iterations: 50
  1. 3x3 SPD System: Tests convergence on sparse symmetric positive definite system
    • Matrix: aa2 (3×3 tridiagonal)
    • Known solution: x2
    • Iterations: 50

Property-Based Tests: 5. Random SPD Systems: QuickCheck property test

  • Generates random SPD matrices via m #^# m
  • Tests convergence for various sizes
  • Iterations: 100

Helper Functions:

  • checkCGS: Runs CGS for n iterations and verifies convergence

    • Takes matrix, RHS, expected solution, and iteration count
    • Returns boolean indicating if residual is near zero
    • Used by all convergence tests

  • prop_cgs: Property test function

    • Tests CGS on random SPD systems
    • Guards against tiny systems (< 2 dimensions)

3. Documentation Updates

COMMENTED_OUT_FUNCTIONS.md

  • Updated to mark CGS functions as ✅ RE-ENABLED
  • Added status notes about tests
  • Updated summary statistics:
    • Originally commented out: 10 functions
    • Re-enabled: 2 (CGS)
    • Still commented: 8

BUILD_AND_TEST.md (New)

Created comprehensive build and test documentation including:

  • Build steps with stack/make commands
  • Expected test outcomes
  • Troubleshooting guide for common issues
  • Performance notes about CGS behavior
  • Verification checklist

4. Code Quality

Code Review:

  • Addressed all review feedback
  • Removed unnecessary MonadIO constraint
  • Improved code comments for clarity
  • Verified test descriptions match test data

Security Scan:

  • Ran CodeQL checker
  • No security vulnerabilities detected

Algorithm Correctness

The CGS implementation follows the standard algorithm from: Y. Saad, "Iterative Methods for Sparse Linear Systems", 2nd ed., 2000

Algorithm Steps:

  1. Initialization:

    • r₀ = b - Ax₀ (initial residual)
    • r̂ = r₀ (fixed shadow residual)
    • p₀ = u₀ = r₀

  2. Iteration:

    • α = (r·r̂) / (Ap·r̂)
    • q = u - αAp
    • x_{j+1} = x + α(u + q)
    • r_{j+1} = r - αA(u + q)
    • β = (r_{j+1}·r̂) / (r·r̂)
    • u_{j+1} = r_{j+1} + βq
    • p_{j+1} = u_{j+1} + β(q + βp)

All mathematical operations in the implementation match this reference.

Files Modified

  1. src/Numeric/LinearAlgebra/Sparse.hs - Uncommented and exported CGS functions
  2. test/LibSpec.hs - Added comprehensive test suite
  3. COMMENTED_OUT_FUNCTIONS.md - Updated documentation
  4. BUILD_AND_TEST.md - Created build/test guide (new file)
  5. IMPLEMENTATION_SUMMARY_CGS.md - This summary (new file)

Remaining Work

Critical (Required Before Merge):

  1. Build Verification: Run stack build to ensure no compilation errors
  2. Test Execution: Run stack test to verify all tests pass
  3. Test Results Review: Verify all CGS tests pass consistently

Post-Merge Considerations:

  1. Integration with <\> operator: Currently, the high-level linSolve0 function is commented out. To fully integrate CGS into the user-facing API, linSolve0 would need to be re-enabled.

  2. Preconditioner Support: For better performance on ill-conditioned systems, consider adding preconditioner support.

  3. Convergence Monitoring: Add optional convergence monitoring/logging for debugging.

  4. Complex Number Support: Test and verify CGS works with Complex Double types.

  5. Performance Benchmarking: Compare CGS performance against other solvers (GMRES, BiCGSTAB, etc.).

Known Limitations

  1. Stability: CGS can exhibit irregular convergence and may fail on ill-conditioned systems
  2. Residual Behavior: Residual may not decrease monotonically
  3. No Preconditioner: Currently no preconditioner support (future enhancement)

Testing Instructions

Quick Test:

stack test --test-arguments "-m CGS"

Full Test Suite:

stack test

Expected Output:

All tests should pass with output similar to:

Numeric.LinearAlgebra.Sparse : CGS (Conjugate Gradient Squared) (Real)
  cgsInit creates initial CGS state
  cgsStep performs one iteration
  CGS converges on 2x2 system
  CGS converges on 3x3 SPD system
QuickCheck properties for CGS:
  prop_cgs : CGS converges for SPD systems
    +++ OK, passed 100 tests.

Conclusion

The CGS solver has been successfully re-enabled with:

  • ✅ Correct algorithm implementation
  • ✅ Comprehensive test coverage (unit, integration, property-based)
  • ✅ Updated documentation
  • ✅ Code review completed
  • ✅ Security scan passed
  • ⏳ Pending: Build and test verification (requires network access)

Once build and tests are verified, the PR is ready for final review and merge.