Linear Algebra and HPC Kernels

This repository contains implementations of numerical linear algebra algorithms and high-performance computing (HPC) kernels:

1. Condition Number Estimation for Symmetric Positive Definite (SPD) matrices
2. Dense Matrix–Vector Multiplication
   • CPU (serial)
   • CPU (OpenMP)
   • GPU (CUDA)
3. Preconditioned Conjugate Gradient (PCG) solver for the 2-D Poisson equation using CSR (Compressed Sparse Row) storage

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1. Condition Number Estimation (SPD Matrices)

This implementation uses the Eigen library.

Dependency:
Eigen (header-only). Clone using
git clone https://gitlab.com/libeigen/eigen.git

Compile:
g++ -O3 -I path/to/eigen cond_number.cpp -o cond_number

Run:
./cond_number /path/to/matrix_file matrix_size

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2. Dense Matrix–Vector Multiplication

CPU (Serial / OpenMP)

Compile (serial):
g++ -O3 Matrix_Vector.cpp -o matvec_cpu

Run:
./matvec_cpu /path/to/A_file /path/to/x_file matrix_size

Compile (OpenMP):
g++ -O3 -fopenmp Matrix_Vector_OMP.cpp -o Exec

Run:
export OMP_NUM_THREADS=#THREADS
./Exec /path/to/A_file /path/to/x_file rwo_size col_size

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GPU (CUDA)

Compile:
nvcc -O3 Matrix_Vector.cu -o matvec_gpu

Run:
./matvec_gpu /path/to/A_file /path/to/x_file row_size col_size threadsPerBlock option

where

• row_size, col_size are the matrix dimensions
• threadsPerBlock specifies the CUDA block size
• option selects the kernel implementation (e.g., 1 or 2)

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3. Preconditioned Conjugate Gradient (PCG)

This implementation solves the 2-D Poisson equation using:

• CSR sparse matrix storage
• Jacobi preconditioning
• Residual-based convergence criteria

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Compile: g++ -O3 cg_solver_pre.cpp -o cg_solver

Run:
./cg_solver /path/to/poisson_dir

4. Prarallel PCG and SSOR

Jacobi Preconditioning in parallel

This implementation solves the 2-D Poisson in CUDA equation using:

• CSR sparse matrix storage
• Jacobi preconditioning
• Residual-based convergence criteria

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Compile: nvcc -O3 -arch=sm_61 --use_fast_math -Xcompiler "-O3" cg_solver_J.cu -o cg_solver_J

Run:
./cg_solver_J /path/to/poisson_dir

SSOR

This implementation solves the 2-D Poisson equation using:

• CSR sparse matrix storage
• SSOR preconditioning
• Residual-based convergence criteria

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Compile: g++ -O3 cg_solver_SSOR.cpp -o cg_solver_SSOR

Run:
./cg_solver_SSOR /path/to/poisson_dir

SSOR in parallel

Compile: nvcc -O3 -arch=sm_61 --use_fast_math -Xcompiler "-O3" cg_solver_J.cu -o cg_solver_SSOR_GPU

Run:
./cg_solver_SSOR_GPU /path/to/poisson_dir