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Vector Operations Documentation

Vector Initialization

Basic Initialization Functions

Function Description Mathematical Expression Example
ca.zeros() Initialize with zeros $\mathbf{v}_i = 0$ ca.vector(5, ca.zeros())
ca.ones() Initialize with ones $\mathbf{v}_i = 1$ ca.vector(5, ca.ones())
ca.constant(c) Initialize with constant $\mathbf{v}_i = c$ ca.vector(5, ca.constant(3.14))

Random Initialization Functions

Function Description Mathematical Expression Example
ca.random(seed?) Uniform [0,1] $\mathbf{v}_i \sim \mathcal{U}(0,1)$ ca.vector(5, ca.random(42))
ca.uniform(min, max, seed?) Uniform [min,max] $\mathbf{v}_i \sim \mathcal{U}(a,b)$ ca.vector(5, ca.uniform(-1, 1))
ca.normal(μ, σ, seed?) Normal distribution $\mathbf{v}_i \sim \mathcal{N}(\mu,\sigma^2)$ ca.vector(5, ca.normal(0, 1))
ca.box_muller(μ, σ, seed?) Box-Muller transform $\mathbf{v}_i \sim \mathcal{N}(\mu,\sigma^2)$ ca.vector(5, ca.box_muller(0, 1))

Mathematical Initialization Functions

Function Description Mathematical Expression Example
ca.sequence(start, step) Arithmetic sequence $\mathbf{v}_i = a + i \cdot d$ ca.vector(5, ca.sequence(0, 2))
ca.arange(start, stop, step) Range sequence $\mathbf{v}_i = a + i \cdot d$ ca.vector(5, ca.arange(0, 10, 2))
ca.sine(freq, phase) Sine wave $\mathbf{v}_i = \sin(2\pi f \cdot i + \phi)$ ca.vector(5, ca.sine(1.0, 0))

Vector Operations

Element-wise Operations

Operation Function Mathematical Expression Example
Multiplication ca.vecmul(a, b) $\mathbf{c}_i = \mathbf{a}_i \cdot \mathbf{b}_i$ c = ca.vecmul(a, b)
Addition ca.vecadd(a, b) $\mathbf{c}_i = \mathbf{a}_i + \mathbf{b}_i$ c = ca.vecadd(a, b)
Subtraction ca.vecsub(a, b) $\mathbf{c}_i = \mathbf{a}_i - \mathbf{b}_i$ c = ca.vecsub(a, b)

Scalar Operations

Operation Function Mathematical Expression Example
Scalar Multiplication ca.vecmul_scalar(a, s) $\mathbf{c}_i = \mathbf{a}_i \cdot s$ c = ca.vecmul_scalar(a, 2.5)
Scalar Addition ca.vecadd_scalar(a, s) $\mathbf{c}_i = \mathbf{a}_i + s$ c = ca.vecadd_scalar(a, 1.0)

Operator Overloading

Operation Operator Mathematical Expression Example
Vector × Vector a * b $\mathbf{c}_i = \mathbf{a}_i \cdot \mathbf{b}_i$ c = a * b
Vector + Vector a + b $\mathbf{c}_i = \mathbf{a}_i + \mathbf{b}_i$ c = a + b
Vector - Vector a - b $\mathbf{c}_i = \mathbf{a}_i - \mathbf{b}_i$ c = a - b
Vector × Scalar a * s $\mathbf{c}_i = \mathbf{a}_i \cdot s$ c = a * 2.5
Vector + Scalar a + s $\mathbf{c}_i = \mathbf{a}_i + s$ c = a + 1.0
Scalar × Vector s * a $\mathbf{c}_i = s \cdot \mathbf{a}_i$ c = 2.5 * a

Usage Examples

Basic Initialization

import cato as ca

# Create vectors with different initializations
zeros = ca.vector(5, ca.zeros())           # [0, 0, 0, 0, 0]
ones = ca.vector(5, ca.ones())             # [1, 1, 1, 1, 1]
constant = ca.vector(5, ca.constant(3.14)) # [3.14, 3.14, 3.14, 3.14, 3.14]
random = ca.vector(5, ca.random(42))       # Random values [0,1]
normal = ca.vector(5, ca.normal(0, 1))     # Normal distribution N(0,1)

Mathematical Sequences

# Arithmetic sequences
seq = ca.vector(5, ca.sequence(0, 2))      # [0, 2, 4, 6, 8]
arange = ca.vector(5, ca.arange(0, 10, 2)) # [0, 2, 4, 6, 8]

# Sine wave
sine = ca.vector(5, ca.sine(1.0, 0))       # [0, 0.95, 0.59, -0.59, -0.95]

Vector Operations

# Create test vectors
a = ca.vector(3, ca.sequence(1, 1))        # [1, 2, 3]
b = ca.vector(3, ca.sequence(2, 1))        # [2, 3, 4]

# Element-wise operations
mul = ca.vecmul(a, b)                      # [2, 6, 12]
add = ca.vecadd(a, b)                      # [3, 5, 7]
sub = ca.vecsub(a, b)                      # [-1, -1, -1]

# Scalar operations
scalar_mul = ca.vecmul_scalar(a, 2.0)      # [2, 4, 6]
scalar_add = ca.vecadd_scalar(a, 1.0)      # [2, 3, 4]

# Using operators
mul_op = a * b                             # [2, 6, 12]
add_op = a + b                             # [3, 5, 7]
scalar_op = a * 2.0                        # [2, 4, 6]

Performance Notes

  • GPU Acceleration: All operations are automatically optimized for CUDA GPUs
  • Memory Management: Automatic CPU ↔ GPU memory transfers
  • Kernel Optimization: Launch parameters are automatically optimized
  • Type Safety: All operations use double precision by default

Requirements

  • Vectors must have the same size for element-wise operations
  • CUDA-compatible GPU required for optimal performance
  • All operations return new vectors (no in-place modification)