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NVIDIA CUDA-X · Scientific Computing

Solve Sparse
Linear Systems
at GPU Speed

Ax = b

NVIDIA cuDSS is the GPU-accelerated Direct Sparse Solver library for engineering and scientific computing. The official replacement for cuSOLVERSp — supporting single-GPU, multi-GPU, and multi-node execution.

↗ Get cuDSS on NVIDIA 📄 Documentation
⚡ Single GPU
Full solver pipeline on one NVIDIA GPU for workstation workloads.
🔗 Multi-GPU
Scale across multiple GPUs for larger sparse systems.
🌐 Multi-Node
MGMN mode via OpenMPI 4.x and NCCL 2.x communication backends.
🔄 Refactorization
Solve sequences of systems with same sparsity at a fraction of cost.
💾 Hybrid Memory
Solve problems larger than GPU memory using automatic hybrid mode.
🏎️ Pascal+
All NVIDIA GPUs Pascal and newer. Optimized for Grace Hopper.
// How It Works

Three-Phase Execution Model

The separation of analysis, factorization, and solve phases enables massive optimization when solving multiple related systems.

01
Analysis
Symbolic analysis of the sparsity pattern. Reordering to minimize fill-in, memory allocation. This phase is computed once and reused across refactorizations. Multithreaded CPU execution in v0.5.0+ for maximum speed.
ReorderingSymbolic factorizationMemory prediction
02
Factorization
Numerical LU, Cholesky, or LDLT factorization executed on the GPU. The computationally intensive phase — massively parallelized on GPU cores. Variable batch support for multiple systems simultaneously.
LU factorizationCholeskyLDLTBatched
03
Solve
Forward and backward triangular solves on GPU. With a factored matrix, solving multiple right-hand sides is extremely fast. For refactorization workflows, skip back to phase 2 with new matrix values.
Forward/backward solveMultiple RHSIterative refinement
// Applications

Powers Engineering & Science

Direct sparse solvers are the computational backbone of simulation software across every engineering discipline.

🏗️
Structural FEM
Finite element method simulations for structural analysis, stress/strain, vibration modes.
🌊
Fluid Dynamics (CFD)
Computational fluid dynamics — pressure-velocity coupling, Navier-Stokes solvers.
⚡
Electromagnetics
EM field simulation, antenna design, radar cross-section, microwave analysis.
🔌
Circuit Simulation
SPICE-like full-chip simulation, transient analysis, power grid analysis.
🚗
Autonomous Driving
Real-time state estimation, sensor fusion, map optimization for self-driving systems.
🏭
Process Simulation
Chemical plant and industrial process modeling with complex reaction networks.
🤖
Physics-Informed AI
PINNs, neural operators, and AI-assisted CAE solving differential equation systems.
📐
LP Optimization
Interior-point method for LP/QP — cuOpt uses cuDSS as its Barrier solver backend.
// Quickstart

Solve Your First Sparse System

cuDSS follows a clean three-step API: create handles, configure matrix descriptors, then call analyze → factor → solve.

1
Create Resources
Initialize the cuDSS handle and matrix/vector descriptors. Define your sparse matrix in CSR format.
2
Analysis Phase
Call cudssSolve(ANALYSIS) to compute reordering and symbolic factorization.
3
Factorization
Call cudssSolve(FACTORIZATION) to compute the numerical LU/Cholesky on GPU.
4
Solve
Call cudssSolve(SOLVE) for triangular solve. Reuse factors for multiple right-hand sides.
cudss_example.c
// Initialize cuDSS handle
cudssHandle_t handle;
cudssCreate(&handle);

// Create matrix descriptor (CSR format)
cudssMatrix_t A;
cudssMatrixCreateCsr(
  &A, n, n, nnz,
  rowOffsets, colIndices, values,
  CUDA_R_64F, CUDSS_MTYPE_GENERAL,
  CUDSS_MVIEW_FULL, CUDSS_BASE_ZERO
);

// Create RHS and solution vectors
cudssMatrix_t b, x;
cudssMatrixCreateDn(&b, n, 1, n, b_vals, CUDA_R_64F, CUDSS_LAYOUT_COL_MAJOR);
cudssMatrixCreateDn(&x, n, 1, n, x_vals, CUDA_R_64F, CUDSS_LAYOUT_COL_MAJOR);

// Three-phase solve: Analysis → Factor → Solve
cudssSolve(handle, CUDSS_PHASE_ANALYSIS, A, x, b);
cudssSolve(handle, CUDSS_PHASE_FACTORIZATION, A, x, b);
cudssSolve(handle, CUDSS_PHASE_SOLVE, A, x, b);

// Cleanup
cudssDestroy(handle);
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