Summary

Complete the triangular solve portion for directly wrapped binaries (MKL and BLIS) to use native LAPACK calls instead of falling back to libblastrampoline. AppleAccelerate was already correctly implemented.

Problem

Previously, the MKL and BLIS LU factorization wrappers were incomplete:

• ✅ Factorization phase (getrf\!): Used native binary calls
• ❌ Triangular solve phase (ldiv\!): Fell back to Julia's ldiv\! → libblastrampoline

This meant that despite having direct native binary access, the solve step still went through the generic Julia Linear Algebra stack.

Solution

MKL (src/mkl.jl)

• Replace ldiv\!(cache.u, factorization, cache.b) with direct getrs\! calls
• Use existing native MKL getrs\! functions that were already implemented but unused
• Handle both square and overdetermined systems correctly

BLIS (ext/LinearSolveBLISExt.jl)

• Replace ldiv\!(cache.u, factorization, cache.b) with direct getrs\! calls
• Use existing native LAPACK getrs\! functions via BLIS that were already implemented but unused
• Add proper error handling with ReturnCode import
• Handle both square and overdetermined systems correctly

AppleAccelerate (src/appleaccelerate.jl)

• ✅ No changes needed - already correctly implemented with native aa_getrs\! calls

Key Changes

MKL solve method:

# Before
y = ldiv\!(cache.u, @get_cacheval(cache, :MKLLUFactorization)[1], cache.b)

# After  
A, info = @get_cacheval(cache, :MKLLUFactorization)
# ... dimension handling ...
getrs\!('N', A.factors, A.ipiv, cache.u; info)

BLIS solve method:

# Before
y = ldiv\!(cache.u, @get_cacheval(cache, :BLISLUFactorization)[1], cache.b)

# After
A, info = @get_cacheval(cache, :BLISLUFactorization)  
# ... dimension handling ...
getrs\!('N', A.factors, A.ipiv, cache.u; info)

Benefits

• Performance: Eliminates libblastrampoline overhead for complete solve process
• Consistency: All three native binaries now use their own LAPACK throughout
• Correctness: Proper handling of both square and overdetermined systems
• Maintainability: Uses existing well-tested getrs\! implementations

Test Results

• ✅ Code compiles successfully without errors
• ✅ No syntax issues detected
• ✅ Standard LU functionality remains intact
• ✅ Native binary loading works correctly when dependencies available

Checklist

• MKL triangular solve uses native MKL getrs\! calls
• BLIS triangular solve uses native LAPACK getrs\! calls via BLIS
• AppleAccelerate confirmed already correct with native aa_getrs\! calls
• Proper error handling for failed factorizations
• Both square and overdetermined system support
• Backward compatibility maintained
• All implementations compile without errors

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