https://docs.rs/kryst

kryst

High-performance Krylov subspace and preconditioned iterative solvers for dense and sparse linear systems, with advanced preconditioning strategies and automated parameter optimization.

Features

Iterative Solvers

• Krylov Methods: CG, PCG, GMRES, FGMRES, BiCGStab, CGS, QMR, TFQMR, MINRES, CGNR
• Direct Methods: LU and QR factorization via PREONLY solver type
• Parallel Support: Shared-memory (Rayon) and distributed-memory (MPI) parallelism

Preconditioners

Basic Preconditioners

• Jacobi: Diagonal scaling preconditioner
• Block Jacobi: Block-wise diagonal preconditioning
• SOR/SSOR: Successive Over-Relaxation methods
• None: No preconditioning (identity)

Incomplete Factorizations

• ILU(0): Zero fill-in incomplete LU factorization
• ILU(k): Incomplete LU with k levels of fill-in
• ILUT: Threshold-based incomplete LU factorization
• ILUTP: ILUT with partial pivoting
• ILUP: Incomplete LU with partial pivoting

Advanced Preconditioners

• Chebyshev: Enhanced polynomial preconditioning with eigenvalue estimation
• AMG: Algebraic Multigrid with configurable smoothing parameters
• ASM: Additive Schwarz Method (domain decomposition)
• Approximate Inverse: SPAI-type approximate inverse preconditioners

Composite Preconditioning

• PC-Chaining: Sequential application of multiple preconditioners via pc_chain option
• Enhanced Chebyshev: Matrix-aware polynomial preconditioning with automatic eigenvalue estimation
• Smoothed AMG: Configurable pre- and post-smoothing parameters (amg_nu_pre, amg_nu_post)

MPI support notes

• MPI-local (per-rank): AMG, Chebyshev, SOR/SSOR, ILU/ILUT/ILUP/ILUTP, Approximate Inverse, LU/QR (dense-direct). These operate on the local block of a distributed matrix.
• Distributed-capable: ASM over DistCsrOp, Block Jacobi on DistCsrOp, and SuperLU_DIST (when the superlu_dist feature is enabled).
• PC-Chaining works on MPI, but each preconditioner in the chain keeps its local vs distributed behavior.

Monitoring & Automation

• Iteration Monitoring: Real-time convergence tracking with IterationMonitor
• Parameter Tuning: Automated optimization with ParameterTuner and grid search
• Data Export: CSV output for convergence analysis with enable_csv_logging()
• Performance Metrics: Comprehensive timing and convergence rate analysis

Scalar Modes

• Real (default): Builds without extra features keep all public APIs monomorphic on f64.
• Complex (--features complex): Internals promote Kryst's scalar alias S to num_complex::Complex64 while the Matrix Market tooling converts boundary data to and from complex storage.

S is the internal scalar alias and R is its real partner. In real builds S = R = f64. In complex builds S = Complex64 and R = f64.

Cargo Features

Feature	Enables	Notes
mpi	MPI communication backend	Requires MPI installed; examples run via mpirun
complex	Complex scalar S	Classical and pipelined GMRES/FGMRES variants are supported
backend-faer	Dense/CSR backends and most PCs	Default feature
backend flags	Direct solvers / matrix backends	e.g. superlu_dist (where available)

Cargo feature summary

• mpi — enable distributed-memory execution via the mpi crate. Optional and independent from Rayon.
• rayon — turn on shared-memory parallel kernels. Combine with -ksp_threads to size the worker pool.
• complex — lift internal kernels to Complex64 while keeping the public API monomorphic on f64 inputs.
• logging — route internal tracing to the log facade for integration with env_logger or similar backends.
• backend-faer + rayon + mpi — supported for distributed runs with parallel local kernels; see docs/matrix_features.md for the expected feature combinations and matrix capabilities.

Latency-aware solver knobs

The Krylov drivers expose command-line options to balance global reductions against additional local work. The most common flags mirror PETSc's -ksp_* options and can be combined with the deterministic reduction feature for reproducible CI runs.

Flag	Default	Effect
`-ksp_cg_variant classic	pipelined`	classic
-ksp_reproducible	false	Enable deterministic reductions (rank-ordered MPI sums and fixed-order local kernels).
-ksp_threads <N>	unset	Request N Rayon workers (requires --features rayon). Ignored in builds without Rayon.
`-ksp_gmres_variant classical	pipelined	sstep[:s]`
-ksp_residual_replacement <iters>	50	Force periodic residual recomputation in pipelined CG to control drift (0 disables).
-ksp_trust_region <radius>	unset	Enable CG trust-region safeguarding with the provided radius.
`-ksp_reorthog never	ifneeded	always`

Rayon tuning

Recommended settings for local kernels:

• -ksp_threads <N> selects the Rayon worker count used by Kryst kernels (shared-memory only).
• KRYST_PAR_CUTOFF=<rows> controls the minimum CSR row count before parallel SpMV is used (default 4096); raise it if you see parallel overhead on small problems.

Legacy -ksp_cg_pipelined remains available as an alias for -ksp_cg_variant pipelined. For bit-for-bit reproducibility, combine -ksp_reproducible with -ksp_threads 1. When Rayon is enabled with more than one worker, runs remain deterministic for a fixed thread count but may differ across thread-count configurations.

Reproducible reductions

When -ksp_reproducible is enabled the solver switches to rank-ordered MPI reductions and fixed-order local kernels. This guarantees bit-for-bit equality between runs that use the same communicator size and Rayon thread count. For strict reproducibility we recommend pinning Rayon to a single thread via -ksp_threads 1 (or the RAYON_NUM_THREADS environment variable); otherwise, results remain deterministic for the configured thread count but may differ between thread-count configurations.

Reproducibility recipe (MPI + Rayon)

Use this configuration when validating deterministic reductions:

RAYON_NUM_THREADS=1 mpirun -n 4 cargo run --example mpi_parallel_demo --features "mpi rayon" -- \
  -ksp_reproducible -ksp_threads 1

Each solver also records the number of global reductions performed in SolveStats::counters.num_global_reductions, making it easy to assert expected latency costs in automated tests.

Hybrid MPI + Rayon scaling recipes

Use these rules of thumb when combining MPI ranks with Rayon threads:

• Throughput-oriented runs: allocate threads per rank so that (MPI ranks) × (threads per rank) matches physical cores. Start with -ksp_threads 2-4 per rank and adjust based on local cache behavior and kernel mix (SpMV vs. ILU/ASM work).
• Reproducibility-oriented runs: keep -ksp_reproducible enabled and fix the thread count per rank (-ksp_threads 1 or RAYON_NUM_THREADS=1). Results remain deterministic for a fixed communicator size and thread count.

Example hybrid runs:

# Throughput-oriented: 4 ranks × 4 threads (16 cores total)
RAYON_NUM_THREADS=4 mpirun -n 4 cargo run --example mpi_parallel_demo --features "mpi rayon" -- \
  -ksp_threads 4

# Reproducible: 4 ranks × 1 thread
RAYON_NUM_THREADS=1 mpirun -n 4 cargo run --example mpi_parallel_demo --features "mpi rayon" -- \
  -ksp_reproducible -ksp_threads 1

For performance studies across MPI-only, Rayon-only, and hybrid builds, run the mpi_rayon_suite benchmark via cargo bench (see scripts/bench_mpi_rayon.sh) to compare ILU and ASM preconditioner workloads on small/medium/large matrices.

Architecture

• PETSc-style API: Unified KSP context for runtime solver selection
• Command-line Options: Complete options database with 50+ parameters
• Trait-based Design: Extensible for custom matrices and preconditioners
• Memory Efficiency: In-place operations and configurable workspace management
• High Performance: Optimized inner kernels with SIMD and parallelization
• Matrix-Free Operators: Shell matrices for callback-based MatVec operations
• Setup Reuse: Two-phase API with preconditioner and workspace recycling
• CSR utilities: zero-copy row_ptr/col_idx/values access and sparse kernels (spgemm, CSR Galerkin triple product)

Installation

Add to your Cargo.toml:

[dependencies]
kryst = "1.0"

Feature Flags

[features]
default = []                  # Opt in to exactly the features you need
rayon = ["dep:rayon", "dep:num_cpus"]
mpi = ["dep:mpi"]
logging = ["dep:log"]
complex = ["dep:num-complex"]
simd = []                     # Auto-tuned std::simd sparse mat-vec kernels
x86_intrinsics = []           # Optional x86_64 gather/prefetch micro-tuning

Enabling the simd feature activates the runtime SpMV planner, which selects between the scalar CSR baseline, a gather-based SIMD kernel, and a SELL-C-σ kernel. Plans are built once per matrix (e.g., during AMG setup) and cached for deterministic, allocation-free application time.

Quick Start

Basic Usage with KspContext (Recommended)

use kryst::prelude::*;
use kryst::matrix::op::DenseOp;
use faer::Mat;
use std::sync::Arc;

// Create a 100×100 test system
let n = 100;
let mat = Mat::<f64>::from_fn(n, n, |i, j| {
    if i == j { 4.0 } else if (i as i32 - j as i32).abs() == 1 { -1.0 } else { 0.0 }
});
let a = Arc::new(DenseOp::<f64>::new(Arc::new(mat)));
let rhs = vec![1.0; n];
let mut solution = vec![0.0; n];

// Configure solver and preconditioner
let mut ksp = KspContext::new();
ksp.set_type(SolverType::Gmres)?
   .set_pc_type(PcType::Jacobi, None)?
   .set_operators(a.clone(), None);
ksp.rtol = 1e-8;
ksp.maxits = 1000;

// Setup once then solve
ksp.setup()?;
let stats = ksp.solve(&rhs, &mut solution)?;
println!(
    "Converged in {} iterations with residual {:.2e}",
    stats.iterations,
    stats.final_residual
);

Explicit Setup and Reuse

Reuse factorization and workspace across multiple solves by calling setup() once:

let mut ksp = KspContext::new();
ksp.set_type(SolverType::Cg)?
   .set_pc_type(PcType::Jacobi, None)?
   .set_operators(a.clone(), None);
ksp.setup()?; // perform factorization and allocate workspace

for rhs in rhs_set.iter() {
    let mut x = vec![0.0; n];
    ksp.solve(rhs, &mut x)?;
}

Advanced Features: Composite Preconditioning

use kryst::context::ksp_context::KspContext;
use kryst::config::options::{KspOptions, PcOptions};

let mut ksp_opts = KspOptions::default();
ksp_opts.ksp_type = Some("cg".into());
let mut pc_opts = PcOptions::default();
pc_opts.pc_chain = Some("jacobi,chebyshev".into());
pc_opts.chebyshev_degree = Some(5);

let mut ksp = KspContext::new();
ksp.set_from_options(&ksp_opts, &pc_opts)?
   .set_operators(a.clone(), None);
ksp.setup()?;
let stats = ksp.solve(&rhs, &mut solution)?;

Enhanced AMG with Smoothing

use kryst::context::ksp_context::{KspContext, SolverType};
use kryst::context::pc_context::PcType;
use kryst::config::options::PcOptions;

let mut pc_opts = PcOptions::default();
pc_opts.amg_levels = Some(4);
pc_opts.amg_strength_threshold = Some(0.25);
pc_opts.amg_nu_pre = Some(2);  // Pre-smoothing steps
pc_opts.amg_nu_post = Some(1); // Post-smoothing steps

let mut ksp = KspContext::new();
ksp.set_type(SolverType::Gmres)?
   .set_pc_type(PcType::Amg, Some(&pc_opts))?
   .set_operators(a.clone(), None);
ksp.setup()?;
let stats = ksp.solve(&rhs, &mut solution)?;

Iteration Monitoring and Analysis

use kryst::{IterationMonitor, ParameterTuner};
use std::time::Duration;

// Monitor convergence behavior
let mut monitor = IterationMonitor::new();
// In practice, integrate monitor with solver iteration callbacks

// Automated parameter tuning
let mut tuner = ParameterTuner::new();
tuner.set_solver_types(vec![SolverType::Cg, SolverType::Gmres]);
tuner.set_pc_types(vec![PcType::Jacobi, PcType::Chebyshev, PcType::Amg]);
tuner.set_tolerances(vec![1e-6, 1e-8]);
tuner.set_max_config_time(Duration::from_secs(30));

let (best_config, all_results) = tuner.tune_parameters(&matrix, &rhs, 5).unwrap();
println!("Best configuration: {:?}", best_config);

Command-line Interface (PETSc-style)

use kryst::config::options::{parse_all_options, KspOptions, PcOptions};
use kryst::context::ksp_context::KspContext;

// Parse command-line options
let args: Vec<String> = std::env::args().collect();
let (ksp_opts, pc_opts) = parse_all_options(&args)?;

// Configure from options
let mut ksp = KspContext::new();
ksp.set_from_all_options(&ksp_opts, &pc_opts)?
   .set_operators(a.clone(), None);
ksp.setup()?;
let stats = ksp.solve(&rhs, &mut solution)?;

Run your program with PETSc-style options:

# Basic solver configuration
./my_program -ksp_type gmres -ksp_rtol 1e-8 -pc_type jacobi

# Direct solvers
./my_program -ksp_type preonly -pc_type lu     # Direct LU solver  
./my_program -ksp_type preonly -pc_type qr     # Direct QR solver

# Advanced preconditioning
./my_program -ksp_type cg -pc_type amg -amg_nu_pre 2 -amg_nu_post 1
./my_program -ksp_type gmres -pc_chain "jacobi,chebyshev" -chebyshev_degree 5

# Show all available options
./my_program -help

Supported Command-line Options

KSP (Krylov Solver) Options

• -ksp_type <solver> - Solver type: cg, pcg, gmres, fgmres, bicgstab, cgs, qmr, tfqmr, minres, cgnr, preonly
• -ksp_rtol <float> - Relative convergence tolerance (default: 1e-5)
• -ksp_atol <float> - Absolute convergence tolerance (default: 1e-50)
• -ksp_dtol <float> - Divergence tolerance (default: 1e5)
• -ksp_max_it <int> - Maximum number of iterations (default: 10000)
• -ksp_gmres_restart <int> - GMRES restart parameter (default: 50)
• -ksp_pc_side <side> - Preconditioning side: left, right, symmetric
• -ksp_reproducible - Enable deterministic reductions; forces rank-ordered MPI sums and stable intra-rank chunking.

PC (Preconditioner) Options

Basic Preconditioner Options

• -pc_type <pc> - Preconditioner type: jacobi, blockjacobi, sor, none

Incomplete Factorization Options

• -pc_type <pc> - ILU variants: ilu0, ilu, ilut, ilutp, ilup
• -pc_ilu_levels <int> - ILU fill levels (default: 0)
• -pc_ilut_drop_tol <float> - ILUT drop tolerance (default: 1e-3)
• -pc_ilut_max_fill <int> - ILUT maximum fill per row (default: 10)

Enhanced Preconditioner Options

• -pc_type chebyshev - Enhanced Chebyshev with eigenvalue estimation
• -chebyshev_degree <int> - Polynomial degree (default: 3)
• -pc_type amg - Algebraic multigrid with smoothing control
• -amg_levels <int> - Number of AMG levels (default: 4)
• -amg_strength_threshold <float> - Strong connection threshold (default: 0.25)
• -amg_nu_pre <int> - Pre-smoothing steps (default: 1)
• -amg_nu_post <int> - Post-smoothing steps (default: 1)

AMG CLI knobs

• -pc_amg - shorthand alias for -pc_type amg.
• -pc_amg_coarsen <rs|hmis|pmis|falgout> - Coarsening strategy (maps to AMGConfig::coarsen_type).
• -pc_amg_interp <classical|direct|multipass|extended|standard> - Interpolation/extended-smoothing variant.
• -pc_amg_smoother <jacobi|gs|gsr|sgs|hgs|l1jacobi|chebyshev> - Smoother applied on each level.
• -pc_amg_smoother_steps <int> and -pc_amg_smoother_omega <float> control smoothing sweeps/relaxation weight.
• -pc_amg_truncation_factor <float> / -pc_amg_interp_maxnnz <int> trim interpolation fill.
• -pc_amg_rap_truncation_factor <float> / -pc_amg_rap_truncation_abs <float> / -pc_amg_rap_maxnnz <int> prune RAP entries.
• -pc_amg_keep_transpose <bool> / -pc_amg_keep_pivot_in_rap <bool> control symmetry-preserving entries.
• -pc_amg_require_spd <bool> / -pc_amg_print_setup <bool> control SPD enforcement and setup printing.

Example AMG invocation:

./solve \
  -pc_amg \
  -pc_amg_levels 6 \
  -pc_amg_strength_threshold 0.25 \
  -pc_amg_coarsen hmis \
  -pc_amg_interp extended \
  -pc_amg_smoother chebyshev \
  -pc_amg_smoother_steps 2 \
  -pc_amg_smoother_omega 0.8 \
  -pc_amg_truncation_factor 0.2 \
  -pc_amg_interp_maxnnz 8 \
  -pc_amg_rap_truncation_factor 0.05 \
  -pc_amg_rap_truncation_abs 0.0 \
  -pc_amg_rap_maxnnz 16 \
  -pc_amg_keep_transpose true \
  -pc_amg_keep_pivot_in_rap true \
  -pc_amg_require_spd true \
  -pc_amg_print_setup true

Composite Preconditioning Options

• -pc_chain <string> - Sequential preconditioner chain (e.g., "jacobi,chebyshev")
• -pc_type asm - Additive Schwarz Method
• -pc_type approxinv - Approximate inverse preconditioner

ILU preconditioners

-pc_type ilu selects Kryst's HYPRE-inspired incomplete LU family (Ilu). -pc_type ilut/-pc_type ilutp run the lighter-weight row-filter ILUT or pivoting ILUTP preconditioners, while -pc_type blockjacobi with -pc_local <ilu|ilut|ilutp> wraps a local ILU variant inside MPI block-Jacobi. Setting -pc_type ilu with -pc_ilu_type ilut runs the canonical ILU threshold factorization; Ilu::create_specialized may route that variant to crate::preconditioner::ilut::Ilut for simplicity/efficiency.

CLI flag	Config field	Notes
`-pc_ilu_type <ilu0	milu0	iluk
-pc_ilu_level_of_fill <int>	IluConfig::level_of_fill	Controls level-of-fill for ILUK (typical 0–5).
-pc_ilu_max_fill_per_row <int>	IluConfig::max_fill_per_row	Per-row fill cap for ILUK/ILUT; 10–50 keeps memory bounded.
-pc_ilu_offdiag_drop_tolerance <float>	IluConfig::offdiag_drop_tolerance	Drop entries outside LU blocks.
-pc_ilu_schur_drop_tolerance <float>	IluConfig::schur_drop_tolerance	For future Schur complements (currently dormant).
`-pc_ilu_triangular_solve <exact	jacobi	gauss_seidel>`
-pc_ilu_lower_jacobi_iters <int> / -pc_ilu_upper_jacobi_iters <int>	Jacobi iteration counts	Only used when the triangular solve is iterative.
-pc_ilu_tolerance <float> / -pc_ilu_max_iterations <int>	Iterative solve controls	Defaults 1e-6 & 1; iterative delivers residual-based refinement.
-pc_ilu_parallel_factorization / -pc_ilu_parallel_trisolve / -pc_ilu_parallel_chunk_size <int>	IluConfig::enable_parallel_*, parallel_chunk_size	Enable experimental rayon paths; chunk size typically 16–256.
-pc_ilut_drop_tol <float>	IluConfig::drop_tolerance (row-filter ILUT)	Simple heuristic ILUT drop threshold (1e-3–1e-6).
-pc_ilut_max_fill <int>	IluConfig::max_fill_per_row (row-filter ILUT)	Limits kept entries per row (10–100).
-pc_ilut_perm_tol <float>	Pivot tolerance for row-filter ILUT	Not used by canonical Ilu but available for the lightweight ILUT preconditioner.
-pc_ilutp_max_fill <int> / -pc_ilutp_drop_tol <float> / -pc_ilutp_perm_tol <float>	Ilutp parameters	Controls density, drop tolerance, and pivoting aggressiveness for ILUTP.

Environment variables mirror the flags: KRYST_PC_ILU_TYPE, KRYST_PC_ILU_LEVEL_OF_FILL, KRYST_PC_ILU_MAX_FILL_PER_ROW, KRYST_PC_ILU_OFFDIAG_DROP_TOL, KRYST_PC_ILU_SCHUR_DROP_TOL, KRYST_PC_ILU_TRI_SOLVE, KRYST_PC_ILU_LOWER_JACOBI_ITERS, KRYST_PC_ILU_UPPER_JACOBI_ITERS, KRYST_PC_ILU_PARALLEL_FACTORIZATION, KRYST_PC_ILU_PARALLEL_TRISOLVE, KRYST_PC_ILU_PARALLEL_CHUNK_SIZE, plus KRYST_PC_ILUT_DROP_TOL, KRYST_PC_ILUT_MAX_FILL, KRYST_PC_ILUT_PERM_TOL, KRYST_PC_ILUTP_MAX_FILL, KRYST_PC_ILUTP_DROP_TOL, and KRYST_PC_ILUTP_PERM_TOL. Command-line flags override environment variables, which in turn override the built-in defaults.

Examples

-pc_type ilu -pc_ilu_type ilu0 -pc_ilu_triangular_solve exact
-pc_type ilu -pc_ilu_type ilut -pc_ilut_drop_tol 1e-5 -pc_ilut_max_fill 50
-pc_type ilutp -pc_ilutp_max_fill 20 -pc_ilutp_drop_tol 1e-4 -pc_ilutp_perm_tol 0.1
-pc_type blockjacobi -pc_local ilu -pc_ilu_type ilu0 -pc_ilu_level_of_fill 1

The first line compares Jacobi vs ILU(0) on examples/poisson_spd_ilu0_vs_jacobi.rs; the second shows ILUT tuning. The third line mirrors the convection–diffusion ILUTP demo (examples/convection_diffusion_ilutp.rs), and the last line is the MPI block-Jacobi + ILU(0) toy from examples/mpi_poisson_block_jacobi_ilu.rs.

Direct Solver Options

• -pc_type lu - Direct LU factorization via SuperLU
• -pc_type qr - Direct QR factorization

Domain Decomposition Options

• -asm_overlap <int> - ASM subdomain overlap (default: 1)
• -asm_type <type> - ASM variant: restrict, interpolate, basic

Usage Examples

# Enhanced Chebyshev preconditioning
-ksp_type cg -pc_type chebyshev -chebyshev_degree 6

# AMG with custom smoothing
-ksp_type gmres -pc_type amg -amg_nu_pre 2 -amg_nu_post 1

# Composite preconditioning (PC-chaining)  
-ksp_type cg -pc_chain "jacobi,chebyshev" -chebyshev_degree 4

# High-accuracy direct solve
-ksp_type preonly -pc_type lu

# BiCGStab with threshold ILU
-ksp_type bicgstab -pc_type ilut -pc_ilut_drop_tol 1e-4

# GMRES with additive Schwarz
-ksp_type gmres -pc_type asm -asm_overlap 2

Monitoring and Automation

Iteration Monitoring

Track solver convergence with real-time monitoring:

use kryst::utils::monitor::IterationMonitor;
use kryst::context::ksp_context::{KspContext, SolverType};
use kryst::context::pc_context::PcType;
use std::sync::{Arc, Mutex};
use std::time::Duration;

// Create and configure monitor
let mut monitor = IterationMonitor::new();
monitor.enable_csv_logging("convergence_history.csv").unwrap();

// Configure solver with monitoring callback
let monitor_ref = Arc::new(Mutex::new(monitor));
let monitor_clone = Arc::clone(&monitor_ref);

let mut ksp = KspContext::new();
ksp.set_type(SolverType::Gmres)?
   .set_pc_type(PcType::Jacobi, None)?
   .set_operators(a.clone(), None);

// Add monitoring callback
ksp.add_monitor(move |iter, residual| {
    if let Ok(mut mon) = monitor_clone.lock() {
        mon.record_iteration(iter, residual, None);
    }
});

// Solve with monitoring
ksp.setup()?;
let stats = ksp.solve(&rhs, &mut solution)?;

// Analyze convergence
if let Ok(mon) = monitor_ref.lock() {
    let convergence_stats = mon.get_statistics();
    println!("Total iterations: {}", convergence_stats.total_iterations);
    println!("Average convergence rate: {:.4}", convergence_stats.avg_convergence_rate);
    println!("Final residual: {:.2e}", convergence_stats.final_residual);
    
    // Check for convergence issues
    if mon.recent_convergence_rate(5).unwrap_or(1.0) > 0.9 {
        println!("Warning: Slow convergence detected");
    }
}

Automated Parameter Tuning

Optimize solver/preconditioner combinations automatically:

use kryst::utils::tuning::{ParameterTuner, ParameterConfig};
use kryst::context::ksp_context::SolverType;
use kryst::context::pc_context::PcType;
use std::time::Duration;

let mut tuner = ParameterTuner::new();

// Configure search space
tuner.set_solver_types(vec![SolverType::Cg, SolverType::Gmres, SolverType::BiCgStab])
     .set_pc_types(vec![PcType::Jacobi, PcType::Chebyshev, PcType::Amg])
     .set_tolerances(vec![1e-6, 1e-8, 1e-10])
     .set_max_config_time(Duration::from_secs(60));

// Add PC-chain configurations for composite preconditioning
tuner.add_pc_chains(vec![
    "jacobi,chebyshev".to_string(),
    "jacobi,ilu0".to_string(),
]);

// Run automated tuning
let (best_config, all_results) = tuner.tune_parameters(&matrix, &rhs, 10).unwrap();

println!("Best configuration found:");
println!("  Solver: {:?}", best_config.solver_type);
println!("  Preconditioner: {:?}", best_config.pc_type);
println!("  Tolerance: {:.2e}", best_config.rtol);
if let Some(chain) = &best_config.pc_chain {
    println!("  PC Chain: {}", chain);
}
println!("  Converged: {}", all_results.iter().find(|r| r.config.solver_type == best_config.solver_type).unwrap().converged);

// Export results for further analysis
tuner.export_results("tuning_results.txt").unwrap();
let summary = tuner.get_summary();
println!("Success rate: {:.1}%", summary.get("convergence_rate").unwrap_or(&0.0) * 100.0);

Advanced Monitoring Features

use kryst::utils::monitor::IterationMonitor;
use std::time::Duration;

let mut monitor = IterationMonitor::new();
monitor.start_solve();

// Record some iterations
monitor.record_iteration(0, 1.0, None);
monitor.record_iteration(1, 0.5, Some(Duration::from_millis(10)));
monitor.record_iteration(2, 0.25, Some(Duration::from_millis(12)));

// Mark convergence
monitor.mark_converged("Relative tolerance achieved");

// Get detailed statistics
let stats = monitor.get_statistics();
println!("Convergence statistics:");
println!("  Total iterations: {}", stats.total_iterations);
println!("  Average convergence rate: {:.4}", stats.avg_convergence_rate);
println!("  Best convergence rate: {:.4}", stats.best_convergence_rate);
println!("  Average iteration time: {:.3}ms", stats.avg_iteration_time.as_secs_f64() * 1000.0);

// Check recent convergence behavior
if let Some(recent_rate) = monitor.recent_convergence_rate(3) {
    println!("Recent convergence rate (last 3 iterations): {:.4}", recent_rate);
}

// Set up real-time monitoring callbacks
let mut ksp = KspContext::new();
ksp.add_monitor(|iter, residual| {
    println!("Iteration {}: residual = {:.3e}", iter, residual);
    
    // Custom monitoring logic
    if iter > 0 && iter % 10 == 0 {
        println!("  Checkpoint: {} iterations completed", iter);
    }
});

Profiling and Performance Analysis

Enable detailed timing and performance information:

[dependencies]
kryst = { version = "1.0", features = ["logging"] }

Run with environment variables for detailed profiling:

# Trace-level logging shows detailed stage timing
RUST_LOG=trace cargo run --features=logging

# Debug-level shows major operations  
RUST_LOG=debug cargo run --features=logging

# Info-level shows high-level progress
RUST_LOG=info cargo run --features=logging

Profiling output includes:

• KSPSetup: Preconditioner setup and workspace allocation timing
• KSPSolve: Complete solve time breakdown
• PCSetup: Individual preconditioner setup timing
• WorkspaceAllocation: Memory allocation timing
• MatVec: Matrix-vector product timing
• PCApply: Preconditioner application timing

Solver Algorithms

Krylov Methods

• CG: Conjugate Gradient for symmetric positive definite systems
• PCG: Preconditioned Conjugate Gradient
• GMRES: Generalized Minimal Residual with restart
• FGMRES: Flexible GMRES for variable preconditioning
• BiCGStab: BiConjugate Gradient Stabilized for nonsymmetric systems
• CGS: Conjugate Gradient Squared
• QMR: Quasi-Minimal Residual method
• TFQMR: Transpose-Free QMR
• MINRES: Minimal Residual for symmetric indefinite systems
• CGNR: Conjugate Gradient on the Normal Equations

Direct Methods

• PREONLY: Single-step direct solve using LU or QR factorization
• Supports both -pc_type lu and -pc_type qr
• Ideal for well-conditioned systems where direct methods are preferred

Preconditioner Details

Basic Preconditioners

• Jacobi: Diagonal scaling M⁻¹ = diag(A)⁻¹
• Block Jacobi: Block-wise diagonal preconditioning with configurable block sizes
• SOR/SSOR: Successive Over-Relaxation with configurable relaxation parameter
• None: Identity preconditioning (no preconditioning)

Incomplete Factorizations

• ILU(0): Zero fill-in incomplete LU factorization
• ILU(k): Incomplete LU with k levels of fill-in
• ILUT: ILU with threshold-based dropping strategy
• ILUTP: ILUT with partial pivoting for numerical stability
• ILUP: Incomplete LU with partial pivoting

Advanced Preconditioners

Enhanced Chebyshev

Enhanced polynomial preconditioning implementation based on eigenvalue estimation:

use kryst::preconditioner::chebyshev::Chebyshev;
use kryst::config::options::PcOptions;
use kryst::context::pc_context::PcType;

// Enhanced Chebyshev with automatic eigenvalue estimation
let mut pc_opts = PcOptions::default();
pc_opts.chebyshev_degree = Some(6);  // Higher degree for better approximation
ksp.set_pc_type(PcType::Chebyshev, Some(&pc_opts))?;

Features:

• Matrix-aware: Automatic eigenvalue bound estimation using power iteration
• Configurable Degree: Polynomial degree optimization (default: 3, range: 1-20)
• Storage Efficient: Reuses matrix storage for eigenvalue computation
• Robust: Handles near-singular matrices with adaptive bounds

Enhanced AMG

Advanced Algebraic Multigrid with configurable smoothing:

use kryst::preconditioner::amg::Amg;
use kryst::config::options::PcOptions;
use kryst::context::pc_context::PcType;

// Enhanced AMG with smoothing control
let mut pc_opts = PcOptions::default();
pc_opts.amg_levels = Some(5);              // Multigrid levels
pc_opts.amg_strength_threshold = Some(0.5); // Strong connection threshold
pc_opts.amg_nu_pre = Some(2);              // Pre-smoothing steps
pc_opts.amg_nu_post = Some(1);             // Post-smoothing steps
ksp.set_pc_type(PcType::Amg, Some(&pc_opts))?;

Features:

• Smoothed Multigrid: Configurable pre- and post-smoothing parameters
• Adaptive Coarsening: Automatic grid hierarchy construction based on strength
• Strength Threshold: Customizable strong connection criteria (default: 0.25)
• Flexible Smoothing: Separate control of pre/post smoothing iterations

Composite Preconditioning

PC-chaining allows sequential application of multiple preconditioners:

use kryst::config::options::{KspOptions, PcOptions};

// Example 1: Jacobi + Chebyshev combination
let mut pc_opts = PcOptions::default();
pc_opts.pc_chain = Some("jacobi,chebyshev".to_string());
pc_opts.chebyshev_degree = Some(4);
ksp.set_from_options(&KspOptions::default(), &pc_opts)?;

// Example 2: Multi-stage preconditioning
let mut pc_opts = PcOptions::default();
pc_opts.pc_chain = Some("jacobi,ilu0,chebyshev".to_string());
ksp.set_from_options(&KspOptions::default(), &pc_opts)?;

// Example 3: Domain decomposition + multigrid
let mut pc_opts = PcOptions::default();
pc_opts.pc_chain = Some("asm,amg".to_string());
pc_opts.amg_nu_pre = Some(1);
ksp.set_from_options(&KspOptions::default(), &pc_opts)?;

Features:

• Flexible Combinations: Mix any preconditioner types in sequence
• Automatic Setup: Transparent handling of composite preconditioner construction
• Parameter Inheritance: Specialized parameters apply to respective stages
• Performance Tuning: Optimize combinations via ParameterTuner

Domain Decomposition

• ASM: Additive Schwarz Method with configurable overlap
• Approximate Inverse: SPAI-type sparse approximate inverse

Performance Features

Parallelization

• Shared Memory: Rayon-based parallel execution for matrix operations and preconditioner application
• Distributed Memory: MPI support for distributed linear algebra operations (via mpi feature)
• SIMD Optimization: Leverages hardware acceleration through optimized inner kernels via faer
• Parallel Preconditioners: Thread-safe preconditioner application with work stealing

Memory Management

• In-place Operations: Minimizes memory allocations during iteration
• Workspace Reuse: Preallocated workspace vectors for Krylov methods
• Block Operations: Efficient cache usage through blocked algorithms
• Sparse Patterns: Memory-efficient storage for sparse matrices and preconditioners

Algorithm Optimizations

• Eigenvalue Estimation: Fast power iteration for Chebyshev eigenvalue bounds
• Adaptive Restart: GMRES restart optimization based on convergence behavior
• Early Termination: Configurable stopping criteria with multiple tolerance options
• Matrix Preprocessing: Reordering and scaling for improved conditioning

Matrix Support

Dense Matrices

• Full support via faer::Mat<T> integration
• Optimized BLAS-level operations
• Support for f32, f64 precision
• Efficient dense matrix-vector products

Sparse Matrices

• Custom CSR format implementation
• Efficient sparse matrix-vector products
• Pattern-based optimization for preconditioners
• Memory-efficient storage with configurable sparsity patterns

Matrix-Free Methods

• Trait-based MatVec interface for custom matrix implementations
• Support for implicit matrix representations
• Easy integration of matrix-free operators
• Efficient for PDE discretizations and other structured problems

Examples and Demonstrations

The library includes comprehensive demonstration programs:

Basic Usage Examples

# Options and CLI interface demonstration
cargo run --example options_demo -- -ksp_type gmres -pc_type jacobi -ksp_rtol 1e-8

# Direct solver usage
cargo run --example dense_direct

# Matrix market file demonstration
cargo run --example matrix_market_demo

Advanced Feature Examples

# Convergence behavior analysis
cargo run --example convergence_demo

# Iteration monitoring demonstration  
cargo run --example monitor -- --features=logging

# HYPRE-style ILU demonstration
cargo run --example hypre_ilu_demo

# MPI parallel examples (requires MPI)
mpirun -n 4 cargo run --example mpi_parallel_demo --features mpi

Note: Matrix Market example files (*.mtx) are excluded from the published crate to stay within size limits. The matrix_market_demo example will auto-generate test data if example files are not found.

Command-line Examples

# Enhanced Chebyshev preconditioning
cargo run --example options_demo -- -ksp_type cg -pc_type chebyshev -chebyshev_degree 6

# AMG with custom smoothing parameters
cargo run --example options_demo -- -ksp_type gmres -pc_type amg -amg_nu_pre 3 -amg_nu_post 2

# Composite preconditioning with PC-chaining
cargo run --example options_demo -- -ksp_type cg -pc_chain "jacobi,chebyshev" -chebyshev_degree 4

# High-precision direct solve
cargo run --example options_demo -- -ksp_type preonly -pc_type lu

# Complex preconditioner combinations
cargo run --example options_demo -- -ksp_type fgmres -pc_type ilut -pc_ilut_drop_tol 1e-5

Benchmarks and Performance

Performance benchmarks are available via:

cargo bench

Benchmark categories include:

• Solver Comparison: GMRES vs BiCGStab vs CG performance on various problems
• Preconditioner Effectiveness: Impact of different preconditioners on convergence
• Direct vs Iterative: Performance comparison for different problem sizes
• Parallel Scaling: Shared-memory (Rayon) and distributed-memory (MPI) performance
• Phase III Features: PC-chaining and enhanced preconditioning performance
• Memory Usage: Workspace allocation and memory efficiency analysis

Sample benchmark results (varies by system and problem):

solver_comparison/gmres    time: 45.2 ms  (convergence: 23 iterations)
solver_comparison/bicgstab time: 38.7 ms  (convergence: 31 iterations)  
solver_comparison/cg       time: 22.1 ms  (convergence: 18 iterations)
pc_effectiveness/jacobi    time: 156 ms   (convergence: 89 iterations)
pc_effectiveness/amg       time: 67.3 ms  (convergence: 12 iterations)
pc_chaining/jacobi+cheby   time: 43.8 ms  (convergence: 15 iterations)

Custom Extensions

Custom Solvers

use kryst::{LinearSolver, MatVec, Preconditioner, SolveStats, KError};

struct MyCustomSolver {
    tolerance: f64,
    max_iterations: usize,
}

impl<M, V> LinearSolver<M, V> for MyCustomSolver 
where 
    M: MatVec<V>,
    V: Clone,
{
    fn solve(
        &mut self, 
        matrix: &M, 
        preconditioner: Option<&dyn Preconditioner<M, V>>, 
        rhs: &V, 
        solution: &mut V
    ) -> Result<SolveStats, KError> {
        // Custom solver implementation
        // Return solve statistics
        Ok(SolveStats {
            iterations: 0,
            residual_norm: 0.0,
            converged: true,
        })
    }
}

Custom Preconditioners

use kryst::{Preconditioner, PcSide, KError};

struct MyCustomPreconditioner {
    // Preconditioner data structures
    factorization: Option<Vec<f64>>,
}

impl<M, V> Preconditioner<M, V> for MyCustomPreconditioner {
    fn setup(&mut self, matrix: &M) -> Result<(), KError> {
        // Preconditioner setup/factorization phase
        // Store factorization data
        Ok(())
    }
    
    fn apply(&self, side: PcSide, x: &V, y: &mut V) -> Result<(), KError> {
        // Apply M⁻¹x → y (or x M⁻¹ → y for right preconditioning)
        match side {
            PcSide::Left => {
                // Left preconditioning: solve Mz = x, return z in y
            },
            PcSide::Right => {
                // Right preconditioning: solve zM = x, return z in y  
            },
        }
        Ok(())
    }
}

Matrix-Free Operators

use kryst::core::traits::MatVec;
use kryst::error::KError;

struct LaplacianOperator {
    n: usize,  // Grid size
    h: f64,    // Grid spacing
}

impl MatVec<Vec<f64>> for LaplacianOperator {
    fn matvec(&self, x: &Vec<f64>, y: &mut Vec<f64>) -> Result<(), KError> {
        // Implement matrix-vector product y = Ax
        // For 1D Laplacian: -u''(x) ≈ -(u[i+1] - 2u[i] + u[i-1])/h²
        
        for i in 0..self.n {
            if i == 0 || i == self.n - 1 {
                y[i] = x[i];  // Boundary conditions
            } else {
                y[i] = (-x[i-1] + 2.0*x[i] - x[i+1]) / (self.h * self.h);
            }
        }
        Ok(())
    }
    
    fn size(&self) -> (usize, usize) {
        (self.n, self.n)
    }
}

// Usage with KspContext
use std::sync::Arc;
let laplacian = Arc::new(LaplacianOperator { n: 1000, h: 0.001 });
let mut ksp = KspContext::new();
ksp.set_type(SolverType::Cg)?
   .set_pc_type(PcType::Jacobi, None)?
   .set_operators(laplacian.clone(), None);

// Can use matrix-free operator directly
let rhs = vec![1.0; laplacian.n];
let mut sol = vec![0.0; laplacian.n];
ksp.setup()?;
let stats = ksp.solve(&rhs, &mut sol)?;

Documentation and Resources

• API Documentation - Complete API reference with examples
• Repository - Source code, issues, and discussions
• Examples Directory - Comprehensive demonstration programs
• Benchmarks - Performance comparison suite
• Phase III/IV Summary - Advanced preconditioning and automation features

Mathematical References

• Saad, Y. (2003). Iterative Methods for Sparse Linear Systems, 2nd Edition. SIAM.
• Barrett, R. et al. (1994). Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM.
• Trefethen, L.N. & Bau, D. (1997). Numerical Linear Algebra. SIAM.
• Briggs, W.L., Henson, V.E. & McCormick, S.F. (2000). A Multigrid Tutorial, 2nd Edition. SIAM.

Software References

• PETSc Documentation: https://petsc.org/release/documentation/
• Trilinos Documentation: https://trilinos.github.io/

Testing and Validation

Run the comprehensive test suite:

# All tests
cargo test

# Specific test categories
cargo test --lib solver
cargo test --lib preconditioner
cargo test --lib context
cargo test --lib utils

# Integration tests
cargo test test_phase_iii_iv_integration
cargo test test_options_integration
cargo test test_preconditioner_integration

# With specific features
cargo test --features "rayon"
cargo test --features "mpi" 
cargo test --features "logging"

# Performance testing
cargo test --release

MPI/Rayon targeted matrix tests

The matrix feature matrix and MPI/Rayon test plan live in docs/matrix_features.md. Use them to validate communicator reductions, distributed SpMV/halo exchange, and Rayon-local kernels for backend-faer + mpi + rayon builds.

Minimal MPI CI recipe

Use the following steps as a minimal MPI validation recipe (local or CI):

mpirun -n 2 cargo test --features "mpi backend-faer"
mpirun -n 2 cargo test --features "mpi rayon backend-faer"

Test Coverage

• Unit Tests: 200+ individual component tests across solvers, preconditioners, and utilities
• Integration Tests: End-to-end validation including monitor integration and parameter tuning
• Options Tests: CLI parsing and configuration validation
• Feature Tests: Advanced functionality validation (PC-chaining, monitoring, tuning)
• Performance Tests: Benchmark validation and regression testing

Migration Guide

From Version 0.x to 1.0

New Features:

• Enhanced Chebyshev preconditioner with eigenvalue estimation
• AMG with configurable pre/post smoothing parameters
• PC-chaining for composite preconditioning
• Iteration monitoring and automated parameter tuning
• Expanded CLI options (50+ parameters)

Breaking Changes:

• None! Version 1.0 maintains full backward compatibility

Recommended Upgrades:

// Old approach
ksp.set_pc_type(PcType::Chebyshev, None)?;

// Enhanced approach (optional)
let mut pc_opts = PcOptions::default();
pc_opts.chebyshev_degree = Some(6);
ksp.set_pc_type(PcType::Chebyshev, Some(&pc_opts))?;

New Monitoring Capabilities:

// Add iteration monitoring
use kryst::utils::monitor::IterationMonitor;
let mut monitor = IterationMonitor::new();
ksp.add_monitor(|iter, residual| {
    println!("Iteration {}: {:.2e}", iter, residual);
});

// Add automated parameter tuning
use kryst::utils::tuning::ParameterTuner;
let mut tuner = ParameterTuner::new();
let (best_config, _) = tuner.tune_parameters(&matrix, &rhs, 5).unwrap();

License

This project is licensed under the MIT License - see the LICENSE file for details.

Contributing

Contributions are welcome! Please feel free to submit a Pull Request. For major changes, please open an issue first to discuss what you would like to change.

Development Setup

1. Clone the repository:

   git clone https://github.com/tmathis720/kryst.git
   cd kryst
   

2. Install Rust (stable toolchain recommended):

   curl --proto '=https' --tlsv1.2 -sSf https://sh.rustup.rs | sh
   

3. Optional: Install MPI for distributed features:

   # Ubuntu/Debian
   sudo apt-get install libopenmpi-dev
   
   # macOS  
   brew install open-mpi
   

4. Run tests and benchmarks:

   cargo test
   cargo bench
   cargo test --features "mpi"  # If MPI is available
   

Developer scripts

• scripts/ci_checks.sh – runs cargo fmt --all -- --check, cargo clippy --all-targets --all-features, and cargo test --all-features.
• scripts/ub_paranoia.sh – executes ASan-enabled tests on the nightly toolchain for the buffer pool and dot engines.
• scripts/miri_reduction.sh – runs the same focused suite under cargo miri (nightly) to catch UB in the unsafe utilities.

Areas for Contribution

High Priority

• GPU Acceleration: CUDA/OpenCL backends for matrix operations
• Additional Solvers: LOBPCG, IDR(s), BiCGStab(l) variants
• Matrix Formats: Coordinate (COO), block sparse (BSR) formats
• Performance: SIMD optimizations, better cache utilization

Medium Priority

• Multigrid Variants: Classical AMG, smoothed aggregation
• Eigenvalue Solvers: Integration with Krylov eigenvalue methods
• Nonlinear Solvers: Newton-Krylov, JFNK methods
• Adaptive Methods: Adaptive restart, dynamic tolerance adjustment

Lower Priority

• Complex Arithmetic: Complex-valued linear systems support
• Mixed Precision: fp16/fp32/fp64 combinations for accuracy/performance tradeoffs
• Advanced I/O: HDF5, NetCDF matrix I/O support
• Visualization: Integration with plotting libraries for convergence analysis

Code Style and Standards

• Follow Rust standard formatting: cargo fmt
• Ensure clippy compliance: cargo clippy
• Add comprehensive tests for new features
• Include benchmark tests for performance-critical code
• Document public APIs with examples
• Follow semantic versioning for releases

Pull Request Process

1. Fork the repository
2. Create a feature branch: git checkout -b feature/amazing-feature
3. Make your changes and add tests
4. Ensure all tests pass: cargo test
5. Run formatting and linting: cargo fmt && cargo clippy
6. Commit your changes: git commit -m 'Add amazing feature'
7. Push to the branch: git push origin feature/amazing-feature
8. Open a Pull Request with a clear description

---

kryst provides a comprehensive, high-performance linear algebra toolkit for the Rust ecosystem, with particular focus on iterative methods for large-scale scientific computing applications. The library combines the mathematical rigor of established numerical libraries like PETSc with the safety and performance characteristics of Rust, making it ideal for research, scientific computing, and production applications requiring robust linear system solvers.