kryst 1.2.1

<!--
    kryst: PETSc-style Krylov solvers and preconditioners for Rust.
    This README describes the main features, usage, and documentation pointers.
-->

# kryst

[![Crates.io](https://img.shields.io/crates/v/kryst.svg)](https://crates.io/crates/kryst)
[![Documentation](https://docs.rs/kryst/badge.svg)](https://docs.rs/kryst)
[![License: MIT](https://img.shields.io/badge/License-MIT-yellow.svg)](https://opensource.org/licenses/MIT)

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`)

### 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

### 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

## Installation

Add to your `Cargo.toml`:

```toml
[dependencies]
kryst = "1.0"
```

### Feature Flags

```toml
[features]
default = ["rayon", "logging"]  # Shared-memory parallelism + monitoring
rayon = ["dep:rayon"]           # Rayon-based parallel execution  
mpi = ["dep:mpi"]              # Distributed-memory parallelism via MPI
logging = ["dep:log"]          # Iteration monitoring and profiling
```

## Quick Start

### Basic Usage with KspContext (Recommended)

```rust
use kryst::context::ksp_context::{KspContext, SolverType};
use kryst::context::pc_context::PcType;
use faer::Mat;

// Create a 100x100 test system
let n = 100;
let matrix = 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 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).unwrap()
   .set_pc_type(PcType::Jacobi).unwrap();
ksp.rtol = 1e-8;
ksp.maxits = 1000;

// Solve the system
ksp.setup(&matrix, n).unwrap();
let stats = ksp.solve(&matrix, &rhs, &mut solution).unwrap();
println!("Converged in {} iterations with residual {:.2e}", 
         stats.iterations, stats.residual_norm);
```

### Advanced Features: Composite Preconditioning

```rust
use kryst::context::ksp_context::{KspContext, SolverType};
use kryst::config::options::PcOptions;

let mut ksp = KspContext::new();
ksp.set_type(SolverType::Cg).unwrap();

// Use PC-chaining for composite preconditioning
let mut pc_opts = PcOptions::default();
pc_opts.pc_chain = Some("jacobi,chebyshev".to_string());
pc_opts.chebyshev_degree = Some(5);
ksp.set_pc_options(pc_opts);

ksp.setup(&matrix, n).unwrap();
let stats = ksp.solve(&matrix, &rhs, &mut solution).unwrap();
```

### Enhanced AMG with Smoothing

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

let mut ksp = KspContext::new();
ksp.set_type(SolverType::Gmres).unwrap()
   .set_pc_type(PcType::Amg).unwrap();

// Configure AMG smoothing parameters
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
ksp.set_pc_options(pc_opts);

ksp.setup(&matrix, n).unwrap();
let stats = ksp.solve(&matrix, &rhs, &mut solution).unwrap();
```

### Iteration Monitoring and Analysis

```rust
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)

```rust
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).unwrap();

// Configure from options  
let mut ksp = KspContext::new();
ksp.set_from_all_options(&ksp_opts, &pc_opts).unwrap();
ksp.setup(&matrix, n).unwrap();
let stats = ksp.solve(&matrix, &rhs, &mut solution).unwrap();
```

Run your program with PETSc-style options:
```bash
# 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-6)
- `-ksp_atol <float>` - Absolute convergence tolerance (default: 1e-12)
- `-ksp_dtol <float>` - Divergence tolerance (default: 1e3)
- `-ksp_max_it <int>` - Maximum number of iterations (default: 1000)
- `-ksp_gmres_restart <int>` - GMRES restart parameter (default: 50)
- `-ksp_pc_side <side>` - Preconditioning side: `left`, `right`, `symmetric`

### 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)

#### 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

#### 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
```bash
# 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:

```rust
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).unwrap()
   .set_pc_type(PcType::Jacobi).unwrap();

// 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(&matrix, n).unwrap();
let stats = ksp.solve(&matrix, &rhs, &mut solution).unwrap();

// 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:

```rust
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

```rust
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:

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

Run with environment variables for detailed profiling:

```bash
# 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:

```rust
use kryst::preconditioner::chebyshev::Chebyshev;
use kryst::config::options::PcOptions;

// Enhanced Chebyshev with automatic eigenvalue estimation
let mut pc_opts = PcOptions::default();
pc_opts.chebyshev_degree = Some(6);  // Higher degree for better approximation
// Lambda bounds are estimated automatically via power iteration
ksp.set_pc_options(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:

```rust
use kryst::preconditioner::amg::Amg;
use kryst::config::options::PcOptions;

// 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_options(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:

```rust
use kryst::config::options::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_pc_options(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_pc_options(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_pc_options(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
```bash
# 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  
```bash
# 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
```bash
# 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:

```bash
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
```rust
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
```rust
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
```rust
use kryst::{MatVec, 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
let laplacian = LaplacianOperator { n: 1000, h: 0.001 };
let mut ksp = KspContext::new();
ksp.set_type(SolverType::Cg).unwrap()
   .set_pc_type(PcType::Jacobi).unwrap();

// Can use matrix-free operator directly
// ksp.setup(&laplacian, 1000).unwrap();
```

## Documentation and Resources

- **[API Documentation](https://docs.rs/kryst)** - Complete API reference with examples
- **[Repository](https://github.com/tmathis720/kryst)** - Source code, issues, and discussions
- **[Examples Directory](https://github.com/tmathis720/kryst/tree/main/examples)** - Comprehensive demonstration programs
- **[Benchmarks](https://github.com/tmathis720/kryst/tree/main/benches)** - Performance comparison suite
- **[Phase III/IV Summary](PHASE_III_IV_SUMMARY.md)** - 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/](https://petsc.org/release/documentation/)
- Trilinos Documentation: [https://trilinos.github.io/](https://trilinos.github.io/)

## Testing and Validation

Run the comprehensive test suite:

```bash
# 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
```

### 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:**
```rust
// Old approach
ksp.set_pc_type(PcType::Chebyshev).unwrap();

// Enhanced approach (optional)
let mut pc_opts = PcOptions::default();
pc_opts.chebyshev_degree = Some(6);
ksp.set_pc_options(pc_opts);
```

**New Monitoring Capabilities:**
```rust
// 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](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:
   ```bash
   git clone https://github.com/tmathis720/kryst.git
   cd kryst
   ```

2. Install Rust (stable toolchain recommended):
   ```bash
   curl --proto '=https' --tlsv1.2 -sSf https://sh.rustup.rs | sh
   ```

3. Optional: Install MPI for distributed features:
   ```bash
   # Ubuntu/Debian
   sudo apt-get install libopenmpi-dev
   
   # macOS  
   brew install open-mpi
   ```

4. Run tests and benchmarks:
   ```bash
   cargo test
   cargo bench
   cargo test --features "mpi"  # If MPI is available
   ```

### 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.