• Stan Functions Reference
  • Overview
  • Built-In Functions
  • 1 Void Functions
    • 1.1 Print statement
    • 1.2 Reject statement
  • 2 Integer-Valued Basic Functions
    • 2.1 Integer-valued arithmetic operators
      • 2.1.1 Binary infix operators
      • 2.1.2 Unary prefix operators
    • 2.2 Absolute functions
    • 2.3 Bound functions
    • 2.4 Size functions
    • 2.5 Casting functions
  • 3 Real-Valued Basic Functions
    • 3.1 Vectorization of real-valued functions
      • 3.1.1 Unary function vectorization
      • 3.1.2 Binary function vectorization
    • 3.2 Mathematical constants
    • 3.3 Special values
    • 3.4 Log probability function
    • 3.5 Logical functions
      • 3.5.1 Comparison operators
      • 3.5.2 Boolean operators
      • 3.5.3 Logical functions
    • 3.6 Real-valued arithmetic operators
      • 3.6.1 Binary infix operators
      • 3.6.2 Unary prefix operators
    • 3.7 Step-like functions
      • 3.7.1 Absolute value functions
      • 3.7.2 Bounds functions
      • 3.7.3 Arithmetic functions
      • 3.7.4 Rounding functions
    • 3.8 Power and logarithm functions
    • 3.9 Trigonometric functions
    • 3.10 Hyperbolic trigonometric functions
    • 3.11 Link functions
    • 3.12 Probability-related functions
      • 3.12.1 Normal cumulative distribution functions
      • 3.12.2 Other probability-related functions
    • 3.13 Combinatorial functions
    • 3.14 Composed functions
    • 3.15 Special functions
  • 4 Complex-Valued Basic Functions
    • 4.1 Complex assignment and promotion
      • 4.1.1 Complex function arguments
    • 4.2 Complex constructors and accessors
      • 4.2.1 Complex constructors
      • 4.2.2 Complex accessors
    • 4.3 Complex arithmetic operators
      • 4.3.1 Unary operators
      • 4.3.2 Binary operators
    • 4.4 Complex comparison operators
    • 4.5 Complex (compound) assignment operators
    • 4.6 Complex special functions
    • 4.7 Complex exponential and power functions
    • 4.8 Complex trigonometric functions
    • 4.9 Complex hyperbolic trigonometric functions
  • 5 Array Operations
    • 5.1 Reductions
      • 5.1.1 Minimum and maximum
      • 5.1.2 Sum, product, and log sum of exp
      • 5.1.3 Sample mean, variance, and standard deviation
      • 5.1.4 Norms
      • 5.1.5 Euclidean distance and squared distance
      • 5.1.6 Quantile
    • 5.2 Array size and dimension function
    • 5.3 Array broadcasting
    • 5.4 Array concatenation
    • 5.5 Sorting functions
    • 5.6 Reversing functions
  • 6 Matrix Operations
    • 6.1 Integer-valued matrix size functions
    • 6.2 Matrix arithmetic operators
      • 6.2.1 Negation prefix operators
      • 6.2.2 Infix matrix operators
      • 6.2.3 Broadcast infix operators
    • 6.3 Transposition operator
    • 6.4 Elementwise functions
    • 6.5 Dot products and specialized products
      • 6.5.1 Specialized products
    • 6.6 Reductions
      • 6.6.1 Log sum of exponents
      • 6.6.2 Minimum and maximum
      • 6.6.3 Sums and products
      • 6.6.4 Sample moments
      • 6.6.5 Quantile
    • 6.7 Broadcast functions
      • 6.7.1 Symmetrization
    • 6.8 Diagonal matrix functions
    • 6.9 Container construction functions
    • 6.10 Slicing and blocking functions
      • 6.10.1 Columns and rows
      • 6.10.2 Block operations
    • 6.11 Matrix concatenation
    • 6.12 Special matrix functions
      • 6.12.1 Softmax
      • 6.12.2 Cumulative sums
    • 6.13 Gaussian Process Covariance Functions
      • 6.13.1 Exponentiated quadratic kernel
      • 6.13.2 Dot product kernel
      • 6.13.3 Exponential kernel
      • 6.13.4 Matern 3/2 kernel
      • 6.13.5 Matern 5/2 kernel
      • 6.13.6 Periodic kernel
    • 6.14 Linear algebra functions and solvers
      • 6.14.1 Matrix division operators and functions
      • 6.14.2 Symmetric positive-definite matrix division functions
      • 6.14.3 Matrix exponential
      • 6.14.4 Matrix power
      • 6.14.5 Linear algebra functions
    • 6.15 Sort functions
    • 6.16 Reverse functions
  • 7 Complex Matrix Operations
    • 7.1 Complex promotion
      • Promotion of complex arguments
      • 7.1.1 Signature selection
      • Signatures for complex functions
      • Generic functions work for complex containers
    • 7.2 Integer-valued complex matrix size functions
    • 7.3 Complex matrix arithmetic operators
      • 7.3.1 Negation prefix operators
      • 7.3.2 Infix complex_matrix operators
      • 7.3.3 Broadcast infix operators
    • 7.4 Complex Transposition Operator
    • 7.5 Complex elementwise functions
    • 7.6 Dot products and specialized products for complex matrices
      • 7.6.1 Specialized products
    • 7.7 Complex reductions
      • 7.7.1 Sums and products
    • 7.8 Vectorized accessor functions
      • 7.8.1 Type “demotion”
      • 7.8.2 Real and imaginary component accessor functions
    • 7.9 Complex broadcast functions
      • 7.9.1 Symmetrization
    • 7.10 Diagonal complex matrix functions
    • 7.11 Slicing and blocking functions for complex matrices
      • 7.11.1 Columns and rows
      • 7.11.2 Block operations
    • 7.12 Complex matrix concatenation
      • 7.12.1 Horizontal concatenation
      • 7.12.2 Vertical concatenation
    • 7.13 Complex special matrix functions
      • 7.13.1 Fast Fourier transforms
      • 7.13.2 Cumulative sums
    • 7.14 Complex linear algebra functions
      • 7.14.1 Complex matrix division operators and functions
      • 7.14.2 Linear algebra functions
    • 7.15 Reverse functions for complex matrices
  • 8 Sparse Matrix Operations
    • 8.1 Compressed row storage
    • 8.2 Conversion functions
      • 8.2.1 Dense to sparse conversion
      • 8.2.2 Sparse to dense conversion
    • 8.3 Sparse matrix arithmetic
      • 8.3.1 Sparse matrix multiplication
  • 9 Mixed Operations
  • 10 Compound Arithmetic and Assignment
    • 10.1 Compound addition and assignment
    • 10.2 Compound subtraction and assignment
    • 10.3 Compound multiplication and assignment
    • 10.4 Compound division and assignment
    • 10.5 Compound elementwise multiplication and assignment
    • 10.6 Compound elementwise division and assignment
  • 11 Higher-Order Functions
    • 11.1 Algebraic equation solvers
      • 11.1.1 Specifying an algebraic equation as a function
      • 11.1.2 Call to the algebraic solver
    • 11.2 Ordinary differential equation (ODE) solvers
      • 11.2.1 Non-stiff solver
      • 11.2.2 Stiff solver
      • 11.2.3 Adjoint solver
      • 11.2.4 ODE system function
      • 11.2.5 Arguments to the ODE solvers
      • 11.2.6 Arguments to the adjoint ODE solver
    • 11.3 Differential-Algebraic equation (DAE) solver
      • 11.3.1 The DAE solver
      • 11.3.2 DAE system function
      • 11.3.3 Arguments to the DAE solver
    • 11.4 1D integrator
      • 11.4.1 Specifying an integrand as a function
      • 11.4.2 Call to the 1D integrator
    • 11.5 Reduce-sum function
      • 11.5.1 Specifying the reduce-sum function
      • 11.5.2 The partial sum function
    • 11.6 Map-rect function
      • 11.6.1 Specifying the mapped function
      • 11.6.2 Rectangular map
  • 12 Deprecated Functions
    • 12.1 multiply_log and binomial_coefficient_log functions
    • 12.2 get_lp() function
    • 12.3 fabs function
    • 12.4 Integer division with operator/
    • 12.5 Real arguments to logical operators operator&&, operator||, and operator!
    • 12.6 integrate_ode_rk45, integrate_ode_adams, integrate_ode_bdf ODE Integrators
      • 12.6.1 Specifying an ordinary differential equation as a function
      • 12.6.2 Non-stiff solver
      • 12.6.3 Stiff solver
      • 12.6.4 Arguments to the ODE solvers
    • 12.7 algebra_solver, algebra_solver_newton algebraic solvers
      • 12.7.1 Specifying an algebraic equation as a function
      • 12.7.2 Call to the algebraic solver
    • 12.8 Exponentiated quadratic covariance functions
  • 13 Removed Functions
  • 14 Conventions for Probability Functions
    • 14.1 Suffix marks type of function
    • 14.2 Argument order and the vertical bar
    • 14.3 Sampling notation
    • 14.4 Finite inputs
    • 14.5 Boundary conditions
    • 14.6 Pseudorandom number generators
      • 14.6.1 Restricted to transformed data and generated quantities
      • 14.6.2 Limited vectorization
    • 14.7 Cumulative distribution functions
    • 14.8 Vectorization
      • 14.8.1 Vectorized function signatures
      • 14.8.2 Evaluating vectorized log probability functions
      • 14.8.3 Evaluating vectorized PRNG functions
  • Discrete Distributions
  • 15 Binary Distributions
    • 15.1 Bernoulli distribution
      • 15.1.1 Probability mass function
      • 15.1.2 Sampling statement
      • 15.1.3 Stan Functions
    • 15.2 Bernoulli distribution, logit parameterization
      • 15.2.1 Probability mass function
      • 15.2.2 Sampling statement
      • 15.2.3 Stan Functions
    • 15.3 Bernoulli-logit generalized linear model (Logistic Regression)
      • 15.3.1 Probability mass function
      • 15.3.2 Sampling statement
      • 15.3.3 Stan Functions
  • 16 Bounded Discrete Distributions
    • 16.1 Binomial distribution
      • 16.1.1 Probability mass function
      • 16.1.2 Log probability mass function
      • 16.1.3 Gradient of log probability mass function
      • 16.1.4 Sampling statement
      • 16.1.5 Stan functions
    • 16.2 Binomial distribution, logit parameterization
      • 16.2.1 Probability mass function
      • 16.2.2 Log probability mass function
      • 16.2.3 Gradient of log probability mass function
      • 16.2.4 Sampling statement
      • 16.2.5 Stan functions
    • 16.3 Beta-binomial distribution
      • 16.3.1 Probability mass function
      • 16.3.2 Sampling statement
      • 16.3.3 Stan functions
    • 16.4 Hypergeometric distribution
      • 16.4.1 Probability mass function
      • 16.4.2 Sampling statement
      • 16.4.3 Stan functions
    • 16.5 Categorical distribution
      • 16.5.1 Probability mass functions
      • 16.5.2 Sampling statement
      • 16.5.3 Sampling statement
      • 16.5.4 Stan functions
    • 16.6 Categorical logit generalized linear model (softmax regression)
      • 16.6.1 Probability mass functions
      • 16.6.2 Sampling statement
      • 16.6.3 Stan functions
    • 16.7 Discrete range distribution
      • 16.7.1 Probability mass functions
      • 16.7.2 Sampling statement
      • 16.7.3 Stan functions
    • 16.8 Ordered logistic distribution
      • 16.8.1 Probability mass function
      • 16.8.2 Sampling statement
      • 16.8.3 Stan functions
    • 16.9 Ordered logistic generalized linear model (ordinal regression)
      • 16.9.1 Probability mass function
      • 16.9.2 Sampling statement
      • 16.9.3 Stan functions
    • 16.10 Ordered probit distribution
      • 16.10.1 Probability mass function
      • 16.10.2 Sampling statement
      • 16.10.3 Stan functions
  • 17 Unbounded Discrete Distributions
    • 17.1 Negative binomial distribution
      • 17.1.1 Probability mass function
      • 17.1.2 Sampling statement
      • 17.1.3 Stan functions
    • 17.2 Negative binomial distribution (alternative parameterization)
      • 17.2.1 Probability mass function
      • 17.2.2 Sampling statement
      • 17.2.3 Stan functions
    • 17.3 Negative binomial distribution (log alternative parameterization)
      • 17.3.1 Sampling statement
      • 17.3.2 Stan functions
    • 17.4 Negative-binomial-2-log generalized linear model (negative binomial regression)
      • 17.4.1 Probability mass function
      • 17.4.2 Sampling statement
      • 17.4.3 Stan functions
    • 17.5 Poisson distribution
      • 17.5.1 Probability mass function
      • 17.5.2 Sampling statement
      • 17.5.3 Stan functions
    • 17.6 Poisson distribution, log parameterization
      • 17.6.1 Probability mass function
      • 17.6.2 Sampling statement
      • 17.6.3 Stan functions
    • 17.7 Poisson-log generalized linear model (Poisson regression)
      • 17.7.1 Probability mass function
      • 17.7.2 Sampling statement
      • 17.7.3 Stan functions
  • 18 Multivariate Discrete Distributions
    • 18.1 Multinomial distribution
      • 18.1.1 Probability mass function
      • 18.1.2 Sampling statement
      • 18.1.3 Stan functions
    • 18.2 Multinomial distribution, logit parameterization
      • 18.2.1 Probability mass function
      • 18.2.2 Sampling statement
      • 18.2.3 Stan functions
  • Continuous Distributions
  • 19 Unbounded Continuous Distributions
    • 19.1 Normal distribution
      • 19.1.1 Probability density function
      • 19.1.2 Sampling statement
      • 19.1.3 Stan functions
      • 19.1.4 Standard normal distribution
      • 19.1.5 Sampling statement
      • 19.1.6 Stan functions
    • 19.2 Normal-id generalized linear model (linear regression)
      • 19.2.1 Probability distribution function
      • 19.2.2 Sampling statement
      • 19.2.3 Stan functions
    • 19.3 Exponentially modified normal distribution
      • 19.3.1 Probability density function
      • 19.3.2 Sampling statement
      • 19.3.3 Stan functions
    • 19.4 Skew normal distribution
      • 19.4.1 Probability density function
      • 19.4.2 Sampling statement
      • 19.4.3 Stan functions
    • 19.5 Student-t distribution
      • 19.5.1 Probability density function
      • 19.5.2 Sampling statement
      • 19.5.3 Stan functions
    • 19.6 Cauchy distribution
      • 19.6.1 Probability density function
      • 19.6.2 Sampling statement
      • 19.6.3 Stan functions
    • 19.7 Double exponential (Laplace) distribution
      • 19.7.1 Probability density function
      • 19.7.2 Sampling statement
      • 19.7.3 Stan functions
    • 19.8 Logistic distribution
      • 19.8.1 Probability density function
      • 19.8.2 Sampling statement
      • 19.8.3 Stan functions
    • 19.9 Gumbel distribution
      • 19.9.1 Probability density function
      • 19.9.2 Sampling statement
      • 19.9.3 Stan functions
    • 19.10 Skew double exponential distribution
      • 19.10.1 Probability density function
      • 19.10.2 Sampling statement
      • 19.10.3 Stan functions
  • 20 Positive Continuous Distributions
    • 20.1 Lognormal distribution
      • 20.1.1 Probability density function
      • 20.1.2 Sampling statement
      • 20.1.3 Stan functions
    • 20.2 Chi-square distribution
      • 20.2.1 Probability density function
      • 20.2.2 Sampling statement
      • 20.2.3 Stan functions
    • 20.3 Inverse chi-square distribution
      • 20.3.1 Probability density function
      • 20.3.2 Sampling statement
      • 20.3.3 Stan functions
    • 20.4 Scaled inverse chi-square distribution
      • 20.4.1 Probability density function
      • 20.4.2 Sampling statement
      • 20.4.3 Stan functions
    • 20.5 Exponential distribution
      • 20.5.1 Probability density function
      • 20.5.2 Sampling statement
      • 20.5.3 Stan functions
    • 20.6 Gamma distribution
      • 20.6.1 Probability density function
      • 20.6.2 Sampling statement
      • 20.6.3 Stan functions
    • 20.7 Inverse gamma Distribution
      • 20.7.1 Probability density function
      • 20.7.2 Sampling statement
      • 20.7.3 Stan functions
    • 20.8 Weibull distribution
      • 20.8.1 Probability density function
      • 20.8.2 Sampling statement
      • 20.8.3 Stan functions
    • 20.9 Frechet distribution
      • 20.9.1 Probability density function
      • 20.9.2 Sampling statement
      • 20.9.3 Stan functions
    • 20.10 Rayleigh distribution
      • 20.10.1 Probability density function
      • 20.10.2 Sampling statement
      • 20.10.3 Stan functions
    • 20.11 Log-logistic distribution
      • 20.11.1 Probability density function
      • 20.11.2 Sampling statement
      • 20.11.3 Stan functions
  • 21 Positive Lower-Bounded Distributions
    • 21.1 Pareto distribution
      • 21.1.1 Probability density function
      • 21.1.2 Sampling statement
      • 21.1.3 Stan functions
    • 21.2 Pareto type 2 distribution
      • 21.2.1 Probability density function
      • 21.2.2 Sampling statement
      • 21.2.3 Stan functions
    • 21.3 Wiener First Passage Time Distribution
      • 21.3.1 Probability density function
      • 21.3.2 Sampling statement
      • 21.3.3 Stan functions
      • 21.3.4 boundaries
  • 22 Continuous Distributions on [0, 1]
    • 22.1 Beta distribution
      • 22.1.1 Probability density function
      • 22.1.2 Sampling statement
      • 22.1.3 Stan functions
    • 22.2 Beta proportion distribution
      • 22.2.1 Probability density function
      • 22.2.2 Sampling statement
      • 22.2.3 Stan functions
  • 23 Circular Distributions
    • 23.1 Von Mises distribution
      • 23.1.1 Probability density function
      • 23.1.2 Sampling statement
      • 23.1.3 Stan functions
      • 23.1.4 Numerical stability
  • 24 Bounded Continuous Distributions
    • 24.1 Uniform distribution
      • 24.1.1 Probability density function
      • 24.1.2 Sampling statement
      • 24.1.3 Stan functions
  • 25 Distributions over Unbounded Vectors
    • 25.1 Multivariate normal distribution
      • 25.1.1 Probability density function
      • 25.1.2 Sampling statement
      • 25.1.3 Stan functions
    • 25.2 Multivariate normal distribution, precision parameterization
      • 25.2.1 Probability density function
      • 25.2.2 Sampling statement
      • 25.2.3 Stan functions
    • 25.3 Multivariate normal distribution, Cholesky parameterization
      • 25.3.1 Probability density function
      • 25.3.2 Sampling statement
      • 25.3.3 Stan functions
    • 25.4 Multivariate Gaussian process distribution
      • 25.4.1 Probability density function
      • 25.4.2 Sampling statement
      • 25.4.3 Stan functions
    • 25.5 Multivariate Gaussian process distribution, Cholesky parameterization
      • 25.5.1 Probability density function
      • 25.5.2 Sampling statement
      • 25.5.3 Stan functions
    • 25.6 Multivariate Student-t distribution
      • 25.6.1 Probability density function
      • 25.6.2 Sampling statement
      • 25.6.3 Stan functions
    • 25.7 Multivariate Student-t distribution, Cholesky parameterization
      • 25.7.1 Probability density function
      • 25.7.2 Sampling statement
      • 25.7.3 Stan functions
    • 25.8 Gaussian dynamic linear models
      • 25.8.1 Sampling statement
      • 25.8.2 Stan functions
  • 26 Simplex Distributions
    • 26.1 Dirichlet distribution
      • 26.1.1 Probability density function
      • 26.1.2 Meaning of Dirichlet parameters
      • 26.1.3 Sampling statement
      • 26.1.4 Stan functions
  • 27 Correlation Matrix Distributions
    • 27.1 LKJ correlation distribution
      • 27.1.1 Probability density function
      • 27.1.2 Sampling statement
      • 27.1.3 Stan functions
    • 27.2 Cholesky LKJ correlation distribution
      • 27.2.1 Probability density function
      • 27.2.2 Sampling statement
      • 27.2.3 Stan functions
  • 28 Covariance Matrix Distributions
    • 28.1 Wishart distribution
      • 28.1.1 Probability density function
      • 28.1.2 Sampling statement
      • 28.1.3 Stan functions
    • 28.2 Wishart distribution, Cholesky Parameterization
      • 28.2.1 Probability density function
      • 28.2.2 Stan functions
    • 28.3 Inverse Wishart distribution
      • 28.3.1 Probability density function
      • 28.3.2 Sampling statement
      • 28.3.3 Stan functions
    • 28.4 Inverse Wishart distribution, Cholesky Parameterization
      • 28.4.1 Probability density function
      • 28.4.2 Stan functions
  • Additional Distributions
  • 29 Hidden Markov Models
    • 29.1 Stan functions
  • Appendix
  • 30 Mathematical Functions
    • 30.1 Beta
    • 30.2 Incomplete beta
    • 30.3 Gamma
    • 30.4 Digamma
  • References

Stan Functions Reference

7 Complex Matrix Operations

  • Complex Promotion
  • Integer-Valued Complex Matrix Size Functions
  • Complex Matrix Arithmetic Operators
  • Complex Transposition Operator
  • Complex Elementwise Functions
  • Dot Products and Specialized Products for Complex Matrices
  • Complex Reductions
  • Vectorized Accessor Functions
  • Complex Broadcast Functions
  • Diagonal Complex Matrix Functions
  • Slicing and Blocking Functions for Complex Matrices
  • Complex Matrix Concatenation
  • Complex Special Matrix Functions
  • Complex Linear Algebra Functions
  • Reverse Functions for Complex Matrices