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Optimizing 2x2 Matrix Multiplication with SIMD and SSE Intrinsics

This document outlines the process of multiplying 2x2 matrices using SIMD algebra and SSE intrinsics in C++. It includes the mathematical breakdown of the matrix calculations and offers guidelines for optimization strategies. Key suggestions involve changing loop orders, unrolling loops, and memory alignment to enhance performance. Additional considerations are provided regarding compiler options and flags to ensure maximized utilization of SSE instructions. Important reminders note the new due date for the assignment.

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Optimizing 2x2 Matrix Multiplication with SIMD and SSE Intrinsics

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  1. Notes on Homework 1

  2. CS267 Lecture 7 2x2 Matrix Multiply • C00 += A00B00 + A01B10 • C10 += A10B00 + A11B10 • C01 += A00B01 + A01B11 • C11 += A10B01 + A11B11 • Rewrite as SIMD algebra • C00_C10 += A00_A10 * B00_B00 • C01_C11 += A00_A10 * B01_B01 • C00_C10 += A01_A11 * B10_B10 • C01_C11 += A01_A11 * B11_B11

  3. Summary of SSE intrinsics • #include <emmintrin.h> • Vector data type: • __m128d Load and store operations: • _mm_load_pd • _mm_store_pd • _mm_loadu_pd • _mm_storeu_pd Load and broadcast across vector • _mm_load1_pd Arithmetic: • _mm_add_pd • _mm_mul_pd

  4. Example: multiplying 2x2 matrices #include <emmintrin.h> c1 = _mm_loadu_pd( C+0*lda ); //load unaligned block in C c2 = _mm_loadu_pd( C+1*lda ); for( inti = 0; i < 2; i++ ) { a = _mm_load_pd( A+i*lda); //load alignedi-th column of A b1 = _mm_load1_pd( B+i+0*lda ); //load i-th row of B b2 = _mm_load1_pd( B+i+1*lda ); c1=_mm_add_pd( c1, _mm_mul_pd( a, b1 ) ); //rank-1 update c2=_mm_add_pd( c2, _mm_mul_pd( a, b2 ) ); } _mm_storeu_pd( C+0*lda, c1 ); //store unaligned block in C _mm_storeu_pd( C+1*lda, c2 );

  5. General suggestions for optimizations • Changing the order of the loops (i, j, k) • changes which matrix stays in memory and the type of product • Blocking for multiple levels (registers or SIMD, L1, L2) • L3 is not important as most matrices will fit and optimizations not likely to bring more benefits • Unrolling the loops • Copy optimizations • will be a large benefit if done for sub-matrix that is kept in memory • careful not to overdo (overhead is high for every level) • Aligning memory and padding • can help with SIMD performance • eliminating special cases and bad performance • Adding SIMD intrinsics • cannot get over 50% without explicit intrinsics or auto-vectorization • Tuning parameters • various block sizes, register sizes (perhaps automate)

  6. Other issues • Checking compilers • options available PGI, PathScale, GNU, Cray • Cray compiler seems to have an issue with explicit intrinsics • Using compiler flags • remember not allowed to use MM specific flags (-matmul) • Checking assembly code for SSE instructions • use the –S flag to see assembly • should have mainly ADDPD & MULPD ops, ADDSD & MULSD are scalar computations • Remember the write-up • want to know what optimizations you tried and what worked and failed • try and have an incremental design and show the performance of multiple iterations • DUE DATE changed, now due Monday Feb 17th at 11:59pm

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