# Matrix Operations

## Integer-valued matrix size functions

int num_elements(vector x)
The total number of elements in the vector x (same as function rows)

Available since 2.5

int num_elements(row_vector x)
The total number of elements in the vector x (same as function cols)

Available since 2.5

int num_elements(matrix x)
The total number of elements in the matrix x. For example, if x is a $$5 \times 3$$ matrix, then num_elements(x) is 15

Available since 2.5

int rows(vector x)
The number of rows in the vector x

Available since 2.0

int rows(row_vector x)
The number of rows in the row vector x, namely 1

Available since 2.0

int rows(matrix x)
The number of rows in the matrix x

Available since 2.0

int cols(vector x)
The number of columns in the vector x, namely 1

Available since 2.0

int cols(row_vector x)
The number of columns in the row vector x

Available since 2.0

int cols(matrix x)
The number of columns in the matrix x

Available since 2.0

int size(vector x)
The size of x, i.e., the number of elements

Available since 2.26

int size(row_vector x)
The size of x, i.e., the number of elements

Available since 2.26

int size(matrix x)
The size of the matrix x. For example, if x is a $$5 \times 3$$ matrix, then size(x) is 15

Available since 2.26

## Matrix arithmetic operators

Stan supports the basic matrix operations using infix, prefix and postfix operations. This section lists the operations supported by Stan along with their argument and result types.

### Negation prefix operators

vector operator-(vector x)
The negation of the vector x.

Available since 2.0

row_vector operator-(row_vector x)
The negation of the row vector x.

Available since 2.0

matrix operator-(matrix x)
The negation of the matrix x.

Available since 2.0

T operator-(T x)
Vectorized version of operator-. If T x is a (possibly nested) array of matrix types, -x is the same shape array where each individual value is negated.

Available since 2.31

### Infix matrix operators

vector operator+(vector x, vector y)
The sum of the vectors x and y.

Available since 2.0

row_vector operator+(row_vector x, row_vector y)
The sum of the row vectors x and y.

Available since 2.0

matrix operator+(matrix x, matrix y)
The sum of the matrices x and y

Available since 2.0

vector operator-(vector x, vector y)
The difference between the vectors x and y.

Available since 2.0

row_vector operator-(row_vector x, row_vector y)
The difference between the row vectors x and y

Available since 2.0

matrix operator-(matrix x, matrix y)
The difference between the matrices x and y

Available since 2.0

vector operator*(real x, vector y)
The product of the scalar x and vector y

Available since 2.0

row_vector operator*(real x, row_vector y)
The product of the scalar x and the row vector y

Available since 2.0

matrix operator*(real x, matrix y)
The product of the scalar x and the matrix y

Available since 2.0

vector operator*(vector x, real y)
The product of the scalar y and vector x

Available since 2.0

matrix operator*(vector x, row_vector y)
The product of the vector x and row vector y

Available since 2.0

row_vector operator*(row_vector x, real y)
The product of the scalar y and row vector x

Available since 2.0

real operator*(row_vector x, vector y)
The product of the row vector x and vector y

Available since 2.0

row_vector operator*(row_vector x, matrix y)
The product of the row vector x and matrix y

Available since 2.0

matrix operator*(matrix x, real y)
The product of the scalar y and matrix x

Available since 2.0

vector operator*(matrix x, vector y)
The product of the matrix x and vector y

Available since 2.0

matrix operator*(matrix x, matrix y)
The product of the matrices x and y

Available since 2.0

vector operator+(vector x, real y)
The result of adding y to every entry in the vector x

Available since 2.0

vector operator+(real x, vector y)
The result of adding x to every entry in the vector y

Available since 2.0

row_vector operator+(row_vector x, real y)
The result of adding y to every entry in the row vector x

Available since 2.0

row_vector operator+(real x, row_vector y)
The result of adding x to every entry in the row vector y

Available since 2.0

matrix operator+(matrix x, real y)
The result of adding y to every entry in the matrix x

Available since 2.0

matrix operator+(real x, matrix y)
The result of adding x to every entry in the matrix y

Available since 2.0

vector operator-(vector x, real y)
The result of subtracting y from every entry in the vector x

Available since 2.0

vector operator-(real x, vector y)
The result of adding x to every entry in the negation of the vector y

Available since 2.0

row_vector operator-(row_vector x, real y)
The result of subtracting y from every entry in the row vector x

Available since 2.0

row_vector operator-(real x, row_vector y)
The result of adding x to every entry in the negation of the row vector y

Available since 2.0

matrix operator-(matrix x, real y)
The result of subtracting y from every entry in the matrix x

Available since 2.0

matrix operator-(real x, matrix y)
The result of adding x to every entry in negation of the matrix y

Available since 2.0

vector operator/(vector x, real y)
The result of dividing each entry in the vector x by y

Available since 2.0

row_vector operator/(row_vector x, real y)
The result of dividing each entry in the row vector x by y

Available since 2.0

matrix operator/(matrix x, real y)
The result of dividing each entry in the matrix x by y

Available since 2.0

## Transposition operator

Matrix transposition is represented using a postfix operator.

matrix operator'(matrix x)
The transpose of the matrix x, written as x'

Available since 2.0

row_vector operator'(vector x)
The transpose of the vector x, written as x'

Available since 2.0

vector operator'(row_vector x)
The transpose of the row vector x, written as x'

Available since 2.0

## Elementwise functions

Elementwise functions apply a function to each element of a vector or matrix, returning a result of the same shape as the argument. There are many functions that are vectorized in addition to the ad hoc cases listed in this section; see section function vectorization for the general cases.

vector operator.*(vector x, vector y)
The elementwise product of y and x

Available since 2.0

row_vector operator.*(row_vector x, row_vector y)
The elementwise product of y and x

Available since 2.0

matrix operator.*(matrix x, matrix y)
The elementwise product of y and x

Available since 2.0

vector operator./(vector x, vector y)
The elementwise quotient of y and x

Available since 2.0

vector operator./(vector x, real y)
The elementwise quotient of y and x

Available since 2.4

vector operator./(real x, vector y)
The elementwise quotient of y and x

Available since 2.4

row_vector operator./(row_vector x, row_vector y)
The elementwise quotient of y and x

Available since 2.0

row_vector operator./(row_vector x, real y)
The elementwise quotient of y and x

Available since 2.4

row_vector operator./(real x, row_vector y)
The elementwise quotient of y and x

Available since 2.4

matrix operator./(matrix x, matrix y)
The elementwise quotient of y and x

Available since 2.0

matrix operator./(matrix x, real y)
The elementwise quotient of y and x

Available since 2.4

matrix operator./(real x, matrix y)
The elementwise quotient of y and x

Available since 2.4

vector operator.^(vector x, vector y)
The elementwise power of y and x

Available since 2.24

vector operator.^(vector x, real y)
The elementwise power of y and x

Available since 2.24

vector operator.^(real x, vector y)
The elementwise power of y and x

Available since 2.24

row_vector operator.^(row_vector x, row_vector y)
The elementwise power of y and x

Available since 2.24

row_vector operator.^(row_vector x, real y)
The elementwise power of y and x

Available since 2.24

row_vector operator.^(real x, row_vector y)
The elementwise power of y and x

Available since 2.24

matrix operator.^(matrix x, matrix y)
The elementwise power of y and x

Available since 2.24

matrix operator.^(matrix x, real y)
The elementwise power of y and x

Available since 2.24

matrix operator.^(real x, matrix y)
The elementwise power of y and x

Available since 2.24

## Dot products and specialized products

real dot_product(vector x, vector y)
The dot product of x and y

Available since 2.0

real dot_product(vector x, row_vector y)
The dot product of x and y

Available since 2.0

real dot_product(row_vector x, vector y)
The dot product of x and y

Available since 2.0

real dot_product(row_vector x, row_vector y)
The dot product of x and y

Available since 2.0

row_vector columns_dot_product(vector x, vector y)
The dot product of the columns of x and y

Available since 2.0

row_vector columns_dot_product(row_vector x, row_vector y)
The dot product of the columns of x and y

Available since 2.0

row_vector columns_dot_product(matrix x, matrix y)
The dot product of the columns of x and y

Available since 2.0

vector rows_dot_product(vector x, vector y)
The dot product of the rows of x and y

Available since 2.0

vector rows_dot_product(row_vector x, row_vector y)
The dot product of the rows of x and y

Available since 2.0

vector rows_dot_product(matrix x, matrix y)
The dot product of the rows of x and y

Available since 2.0

real dot_self(vector x)
The dot product of the vector x with itself

Available since 2.0

real dot_self(row_vector x)
The dot product of the row vector x with itself

Available since 2.0

row_vector columns_dot_self(vector x)
The dot product of the columns of x with themselves

Available since 2.0

row_vector columns_dot_self(row_vector x)
The dot product of the columns of x with themselves

Available since 2.0

row_vector columns_dot_self(matrix x)
The dot product of the columns of x with themselves

Available since 2.0

vector rows_dot_self(vector x)
The dot product of the rows of x with themselves

Available since 2.0

vector rows_dot_self(row_vector x)
The dot product of the rows of x with themselves

Available since 2.0

vector rows_dot_self(matrix x)
The dot product of the rows of x with themselves

Available since 2.0

### Specialized products

matrix tcrossprod(matrix x)
The product of x postmultiplied by its own transpose, similar to the tcrossprod(x) function in R. The result is a symmetric matrix $$\text{x}\,\text{x}^{\top}$$.

Available since 2.0

matrix crossprod(matrix x)
The product of x premultiplied by its own transpose, similar to the crossprod(x) function in R. The result is a symmetric matrix $$\text{x}^{\top}\,\text{x}$$.

Available since 2.0

The following functions all provide shorthand forms for common expressions, which are also much more efficient.

matrix quad_form(matrix A, matrix B)
The quadratic form, i.e., B' * A * B.

Available since 2.0

real quad_form(matrix A, vector B)
The quadratic form, i.e., B' * A * B.

Available since 2.0

matrix quad_form_diag(matrix m, vector v)
The quadratic form using the column vector v as a diagonal matrix, i.e., diag_matrix(v) * m * diag_matrix(v).

Available since 2.3

matrix quad_form_diag(matrix m, row_vector rv)
The quadratic form using the row vector rv as a diagonal matrix, i.e., diag_matrix(rv) * m * diag_matrix(rv).

Available since 2.3

matrix quad_form_sym(matrix A, matrix B)
Similarly to quad_form, gives B' * A * B, but additionally checks if A is symmetric and ensures that the result is also symmetric.

Available since 2.3

real quad_form_sym(matrix A, vector B)
Similarly to quad_form, gives B' * A * B, but additionally checks if A is symmetric and ensures that the result is also symmetric.

Available since 2.3

real trace_quad_form(matrix A, matrix B)
The trace of the quadratic form, i.e., trace(B' * A * B).

Available since 2.0

real trace_gen_quad_form(matrix D,matrix A, matrix B)
The trace of a generalized quadratic form, i.e., trace(D * B' * A * B).

Available since 2.0

matrix multiply_lower_tri_self_transpose(matrix x)
The product of the lower triangular portion of x (including the diagonal) times its own transpose; that is, if L is a matrix of the same dimensions as x with L(m,n) equal to x(m,n) for $$\text{n} \leq \text{m}$$ and L(m,n) equal to 0 if $$\text{n} > \text{m}$$, the result is the symmetric matrix $$\text{L}\,\text{L}^{\top}$$. This is a specialization of tcrossprod(x) for lower-triangular matrices. The input matrix does not need to be square.

Available since 2.0

matrix diag_pre_multiply(vector v, matrix m)
Return the product of the diagonal matrix formed from the vector v and the matrix m, i.e., diag_matrix(v) * m.

Available since 2.0

matrix diag_pre_multiply(row_vector rv, matrix m)
Return the product of the diagonal matrix formed from the vector rv and the matrix m, i.e., diag_matrix(rv) * m.

Available since 2.0

matrix diag_post_multiply(matrix m, vector v)
Return the product of the matrix m and the diagonal matrix formed from the vector v, i.e., m * diag_matrix(v).

Available since 2.0

matrix diag_post_multiply(matrix m, row_vector rv)
Return the product of the matrix m and the diagonal matrix formed from the the row vector rv, i.e., m * diag_matrix(rv).

Available since 2.0

## Reductions

### Log sum of exponents

real log_sum_exp(vector x)
The natural logarithm of the sum of the exponentials of the elements in x

Available since 2.0

real log_sum_exp(row_vector x)
The natural logarithm of the sum of the exponentials of the elements in x

Available since 2.0

real log_sum_exp(matrix x)
The natural logarithm of the sum of the exponentials of the elements in x

Available since 2.0

### Minimum and maximum

real min(vector x)
The minimum value in x, or $$+\infty$$ if x is empty

Available since 2.0

real min(row_vector x)
The minimum value in x, or $$+\infty$$ if x is empty

Available since 2.0

real min(matrix x)
The minimum value in x, or $$+\infty$$ if x is empty

Available since 2.0

real max(vector x)
The maximum value in x, or $$-\infty$$ if x is empty

Available since 2.0

real max(row_vector x)
The maximum value in x, or $$-\infty$$ if x is empty

Available since 2.0

real max(matrix x)
The maximum value in x, or $$-\infty$$ if x is empty

Available since 2.0

### Sums and products

real sum(vector x)
The sum of the values in x, or 0 if x is empty

Available since 2.0

real sum(row_vector x)
The sum of the values in x, or 0 if x is empty

Available since 2.0

real sum(matrix x)
The sum of the values in x, or 0 if x is empty

Available since 2.0

real prod(vector x)
The product of the values in x, or 1 if x is empty

Available since 2.0

real prod(row_vector x)
The product of the values in x, or 1 if x is empty

Available since 2.0

real prod(matrix x)
The product of the values in x, or 1 if x is empty

Available since 2.0

### Sample moments

Full definitions are provided for sample moments in section array reductions.

real mean(vector x)
The sample mean of the values in x; see section array reductions for details.

Available since 2.0

real mean(row_vector x)
The sample mean of the values in x; see section array reductions for details.

Available since 2.0

real mean(matrix x)
The sample mean of the values in x; see section array reductions for details.

Available since 2.0

real variance(vector x)
The sample variance of the values in x; see section array reductions for details.

Available since 2.0

real variance(row_vector x)
The sample variance of the values in x; see section array reductions for details.

Available since 2.0

real variance(matrix x)
The sample variance of the values in x; see section array reductions for details.

Available since 2.0

real sd(vector x)
The sample standard deviation of the values in x; see section array reductions for details.

Available since 2.0

real sd(row_vector x)
The sample standard deviation of the values in x; see section array reductions for details.

Available since 2.0

real sd(matrix x)
The sample standard deviation of the values in x; see section array reductions for details.

Available since 2.0

### Quantile

Produces sample quantiles corresponding to the given probabilities. The smallest observation corresponds to a probability of 0 and the largest to a probability of 1.

Implements algorithm 7 from Hyndman, R. J. and Fan, Y., Sample quantiles in Statistical Packages (R’s default quantile function).

real quantile(data vector x, data real p)
The p-th quantile of x

Available since 2.27

array[] real quantile(data vector x, data array[] real p)
An array containing the quantiles of x given by the array of probabilities p

Available since 2.27

real quantile(data row_vector x, data real p)
The p-th quantile of x

Available since 2.27

array[] real quantile(data row_vector x, data array[] real p)
An array containing the quantiles of x given by the array of probabilities p

Available since 2.27

The following broadcast functions allow vectors, row vectors and matrices to be created by copying a single element into all of their cells. Matrices may also be created by stacking copies of row vectors vertically or stacking copies of column vectors horizontally.

vector rep_vector(real x, int m)
Return the size m (column) vector consisting of copies of x.

Available since 2.0

row_vector rep_row_vector(real x, int n)
Return the size n row vector consisting of copies of x.

Available since 2.0

matrix rep_matrix(real x, int m, int n)
Return the m by n matrix consisting of copies of x.

Available since 2.0

matrix rep_matrix(vector v, int n)
Return the m by n matrix consisting of n copies of the (column) vector v of size m.

Available since 2.0

matrix rep_matrix(row_vector rv, int m)
Return the m by n matrix consisting of m copies of the row vector rv of size n.

Available since 2.0

Unlike the situation with array broadcasting (see section array broadcasting), where there is a distinction between integer and real arguments, the following two statements produce the same result for vector broadcasting; row vector and matrix broadcasting behave similarly.

 vector[3] x;
x = rep_vector(1, 3);
x = rep_vector(1.0, 3);

There are no integer vector or matrix types, so integer values are automatically promoted.

### Symmetrization

matrix symmetrize_from_lower_tri(matrix A)

Construct a symmetric matrix from the lower triangle of A.

Available since 2.26

## Diagonal matrix functions

matrix add_diag(matrix m, row_vector d)
Add row_vector d to the diagonal of matrix m.

Available since 2.21

matrix add_diag(matrix m, vector d)
Add vector d to the diagonal of matrix m.

Available since 2.21

matrix add_diag(matrix m, real d)
Add scalar d to every diagonal element of matrix m.

Available since 2.21

vector diagonal(matrix x)
The diagonal of the matrix x

Available since 2.0

matrix diag_matrix(vector x)
The diagonal matrix with diagonal x

Available since 2.0

Although the diag_matrix function is available, it is unlikely to ever show up in an efficient Stan program. For example, rather than converting a diagonal to a full matrix for use as a covariance matrix,

 y ~ multi_normal(mu, diag_matrix(square(sigma)));

it is much more efficient to just use a univariate normal, which produces the same density,

 y ~ normal(mu, sigma);

Rather than writing m * diag_matrix(v) where m is a matrix and v is a vector, it is much more efficient to write diag_post_multiply(m, v) (and similarly for pre-multiplication). By the same token, it is better to use quad_form_diag(m, v) rather than quad_form(m, diag_matrix(v)).

matrix identity_matrix(int k)
Create an identity matrix of size $$k \times k$$

Available since 2.26

## Container construction functions

array[] real linspaced_array(int n, data real lower, data real upper)
Create a real array of length n of equidistantly-spaced elements between lower and upper

Available since 2.24

array[] int linspaced_int_array(int n, int lower, int upper)
Create a regularly spaced, increasing integer array of length n between lower and upper, inclusively. If (upper - lower) / (n - 1) is less than one, repeat each output (n - 1) / (upper - lower) times. If neither (upper - lower) / (n - 1) or (n - 1) / (upper - lower) are integers, upper is reduced until one of these is true.

Available since 2.26

vector linspaced_vector(int n, data real lower, data real upper)
Create an n-dimensional vector of equidistantly-spaced elements between lower and upper

Available since 2.24

row_vector linspaced_row_vector(int n, data real lower, data real upper)
Create an n-dimensional row-vector of equidistantly-spaced elements between lower and upper

Available since 2.24

array[] int one_hot_int_array(int n, int k)
Create a one-hot encoded int array of length n with array[k] = 1

Available since 2.26

array[] real one_hot_array(int n, int k)
Create a one-hot encoded real array of length n with array[k] = 1

Available since 2.24

vector one_hot_vector(int n, int k)
Create an n-dimensional one-hot encoded vector with vector[k] = 1

Available since 2.24

row_vector one_hot_row_vector(int n, int k)
Create an n-dimensional one-hot encoded row-vector with row_vector[k] = 1

Available since 2.24

array[] int ones_int_array(int n)
Create an int array of length n of all ones

Available since 2.26

array[] real ones_array(int n)
Create a real array of length n of all ones

Available since 2.26

vector ones_vector(int n)
Create an n-dimensional vector of all ones

Available since 2.26

row_vector ones_row_vector(int n)
Create an n-dimensional row-vector of all ones

Available since 2.26

array[] int zeros_int_array(int n)
Create an int array of length n of all zeros

Available since 2.26

array[] real zeros_array(int n)
Create a real array of length n of all zeros

Available since 2.24

vector zeros_vector(int n)
Create an n-dimensional vector of all zeros

Available since 2.24

row_vector zeros_row_vector(int n)
Create an n-dimensional row-vector of all zeros

Available since 2.24

vector uniform_simplex(int n)
Create an n-dimensional simplex with elements vector[i] = 1 / n for all $$i \in 1, \dots, n$$

Available since 2.24

## Slicing and blocking functions

Stan provides several functions for generating slices or blocks or diagonal entries for matrices.

### Columns and rows

vector col(matrix x, int n)
The n-th column of matrix x

Available since 2.0

row_vector row(matrix x, int m)
The m-th row of matrix x

Available since 2.0

The row function is special in that it may be used as an lvalue in an assignment statement (i.e., something to which a value may be assigned). The row function is also special in that the indexing notation x[m] is just an alternative way of writing row(x,m). The col function may not, be used as an lvalue, nor is there an indexing based shorthand for it.

### Block operations

#### Matrix slicing operations

Block operations may be used to extract a sub-block of a matrix.

matrix block(matrix x, int i, int j, int n_rows, int n_cols)
Return the submatrix of x that starts at row i and column j and extends n_rows rows and n_cols columns.

Available since 2.0

The sub-row and sub-column operations may be used to extract a slice of row or column from a matrix

vector sub_col(matrix x, int i, int j, int n_rows)
Return the sub-column of x that starts at row i and column j and extends n_rows rows and 1 column.

Available since 2.0

row_vector sub_row(matrix x, int i, int j, int n_cols)
Return the sub-row of x that starts at row i and column j and extends 1 row and n_cols columns.

Available since 2.0

#### Vector and array slicing operations

The head operation extracts the first $$n$$ elements of a vector and the tail operation the last. The segment operation extracts an arbitrary subvector.

vector head(vector v, int n)
Return the vector consisting of the first n elements of v.

Available since 2.0

row_vector head(row_vector rv, int n)
Return the row vector consisting of the first n elements of rv.

Available since 2.0

array[] T head(array[] T sv, int n)
Return the array consisting of the first n elements of sv; applies to up to three-dimensional arrays containing any type of elements T.

Available since 2.0

vector tail(vector v, int n)
Return the vector consisting of the last n elements of v.

Available since 2.0

row_vector tail(row_vector rv, int n)
Return the row vector consisting of the last n elements of rv.

Available since 2.0

array[] T tail(array[] T sv, int n)
Return the array consisting of the last n elements of sv; applies to up to three-dimensional arrays containing any type of elements T.

Available since 2.0

vector segment(vector v, int i, int n)
Return the vector consisting of the n elements of v starting at i; i.e., elements i through through i + n - 1.

Available since 2.0

row_vector segment(row_vector rv, int i, int n)
Return the row vector consisting of the n elements of rv starting at i; i.e., elements i through through i + n - 1.

Available since 2.10

array[] T segment(array[] T sv, int i, int n)
Return the array consisting of the n elements of sv starting at i; i.e., elements i through through i + n - 1. Applies to up to three-dimensional arrays containing any type of elements T.

Available since 2.0

## Matrix concatenation

Stan’s matrix concatenation operations append_col and append_row are like the operations cbind and rbind in R.

#### Horizontal concatenation

matrix append_col(matrix x, matrix y)
Combine matrices x and y by column. The matrices must have the same number of rows.

Available since 2.5

matrix append_col(matrix x, vector y)
Combine matrix x and vector y by column. The matrix and the vector must have the same number of rows.

Available since 2.5

matrix append_col(vector x, matrix y)
Combine vector x and matrix y by column. The vector and the matrix must have the same number of rows.

Available since 2.5

matrix append_col(vector x, vector y)
Combine vectors x and y by column. The vectors must have the same number of rows.

Available since 2.5

row_vector append_col(row_vector x, row_vector y)
Combine row vectors x and y of any size into another row vector by appending y to the end of x.

Available since 2.5

row_vector append_col(real x, row_vector y)
Append x to the front of y, returning another row vector.

Available since 2.12

row_vector append_col(row_vector x, real y)
Append y to the end of x, returning another row vector.

Available since 2.12

#### Vertical concatenation

matrix append_row(matrix x, matrix y)
Combine matrices x and y by row. The matrices must have the same number of columns.

Available since 2.5

matrix append_row(matrix x, row_vector y)
Combine matrix x and row vector y by row. The matrix and the row vector must have the same number of columns.

Available since 2.5

matrix append_row(row_vector x, matrix y)
Combine row vector x and matrix y by row. The row vector and the matrix must have the same number of columns.

Available since 2.5

matrix append_row(row_vector x, row_vector y)
Combine row vectors x and y by row. The row vectors must have the same number of columns.

Available since 2.5

vector append_row(vector x, vector y)
Concatenate vectors x and y of any size into another vector.

Available since 2.5

vector append_row(real x, vector y)
Append x to the top of y, returning another vector.

Available since 2.12

vector append_row(vector x, real y)
Append y to the bottom of x, returning another vector.

Available since 2.12

## Special matrix functions

### Softmax

The softmax function maps1 $$y \in \mathbb{R}^K$$ to the $$K$$-simplex by $\begin{equation*} \text{softmax}(y) = \frac{\exp(y)} {\sum_{k=1}^K \exp(y_k)}, \end{equation*}$ where $$\exp(y)$$ is the componentwise exponentiation of $$y$$. Softmax is usually calculated on the log scale, $\begin{eqnarray*} \log \text{softmax}(y) & = & \ y - \log \sum_{k=1}^K \exp(y_k) \\[4pt] & = & y - \mathrm{log\_sum\_exp}(y). \end{eqnarray*}$ where the vector $$y$$ minus the scalar $$\mathrm{log\_sum\_exp}(y)$$ subtracts the scalar from each component of $$y$$.

Stan provides the following functions for softmax and its log.

vector softmax(vector x)
The softmax of x

Available since 2.0

vector log_softmax(vector x)
The natural logarithm of the softmax of x

Available since 2.0

### Cumulative sums

The cumulative sum of a sequence $$x_1,\ldots,x_N$$ is the sequence $$y_1,\ldots,y_N$$, where $\begin{equation*} y_n = \sum_{m = 1}^{n} x_m. \end{equation*}$

array[] int cumulative_sum(array[] int x)
The cumulative sum of x

Available since 2.30

array[] real cumulative_sum(array[] real x)
The cumulative sum of x

Available since 2.0

vector cumulative_sum(vector v)
The cumulative sum of v

Available since 2.0

row_vector cumulative_sum(row_vector rv)
The cumulative sum of rv

Available since 2.0

## Gaussian Process Covariance Functions

The Gaussian process covariance functions compute the covariance between observations in an input data set or the cross-covariance between two input data sets.

For one dimensional GPs, the input data sets are arrays of scalars. The covariance matrix is given by $$K_{ij} = k(x_i, x_j)$$ (where $$x_i$$ is the $$i^{th}$$ element of the array $$x$$) and the cross-covariance is given by $$K_{ij} = k(x_i, y_j)$$.

For multi-dimensional GPs, the input data sets are arrays of vectors. The covariance matrix is given by $$K_{ij} = k(\mathbf{x}_i, \mathbf{x}_j)$$ (where $$\mathbf{x}_i$$ is the $$i^{th}$$ vector in the array $$x$$) and the cross-covariance is given by $$K_{ij} = k(\mathbf{x}_i, \mathbf{y}_j)$$.

With magnitude $$\sigma$$ and length scale $$l$$, the exponentiated quadratic kernel is:

$k(\mathbf{x}_i, \mathbf{x}_j) = \sigma^2 \exp \left( -\frac{|\mathbf{x}_i - \mathbf{x}_j|^2}{2l^2} \right)$

matrix gp_exp_quad_cov(array[] real x, real sigma, real length_scale)

Gaussian process covariance with exponentiated quadratic kernel in one dimension.

Available since 2.20

matrix gp_exp_quad_cov(array[] real x1, array[] real x2, real sigma, real length_scale)

Gaussian process cross-covariance of x1 and x2 with exponentiated quadratic kernel in one dimension.

Available since 2.20

matrix gp_exp_quad_cov(vectors x, real sigma, real length_scale)

Gaussian process covariance with exponentiated quadratic kernel in multiple dimensions.

Available since 2.20

matrix gp_exp_quad_cov(vectors x, real sigma, array[] real length_scale)

Gaussian process covariance with exponentiated quadratic kernel in multiple dimensions with a length scale for each dimension.

Available since 2.20

matrix gp_exp_quad_cov(vectors x1, vectors x2, real sigma, real length_scale)

Gaussian process cross-covariance of x1 and x2 with exponentiated quadratic kernel in multiple dimensions.

Available since 2.20

matrix gp_exp_quad_cov(vectors x1, vectors x2, real sigma, array[] real length_scale)

Gaussian process cross-covariance of x1 and x2 with exponentiated quadratic kernel in multiple dimensions with a length scale for each dimension.

Available since 2.20

### Dot product kernel

With bias $$\sigma_0$$ the dot product kernel is:

$k(\mathbf{x}_i, \mathbf{x}_j) = \sigma_0^2 + \mathbf{x}_i^T \mathbf{x}_j$

matrix gp_dot_prod_cov(array[] real x, real sigma)

Gaussian process covariance with dot product kernel in one dimension.

Available since 2.20

matrix gp_dot_prod_cov(array[] real x1, array[] real x2, real sigma)

Gaussian process cross-covariance of x1 and x2 with dot product kernel in one dimension.

Available since 2.20

matrix gp_dot_prod_cov(vectors x, real sigma)

Gaussian process covariance with dot product kernel in multiple dimensions.

Available since 2.20

matrix gp_dot_prod_cov(vectors x1, vectors x2, real sigma)

Gaussian process cross-covariance of x1 and x2 with dot product kernel in multiple dimensions.

Available since 2.20

### Exponential kernel

With magnitude $$\sigma$$ and length scale $$l$$, the exponential kernel is:

$k(\mathbf{x}_i, \mathbf{x}_j) = \sigma^2 \exp \left( -\frac{|\mathbf{x}_i - \mathbf{x}_j|}{l} \right)$

matrix gp_exponential_cov(array[] real x, real sigma, real length_scale)

Gaussian process covariance with exponential kernel in one dimension.

Available since 2.20

matrix gp_exponential_cov(array[] real x1, array[] real x2, real sigma, real length_scale)

Gaussian process cross-covariance of x1 and x2 with exponential kernel in one dimension.

Available since 2.20

matrix gp_exponential_cov(vectors x, real sigma, real length_scale)

Gaussian process covariance with exponential kernel in multiple dimensions.

Available since 2.20

matrix gp_exponential_cov(vectors x, real sigma, array[] real length_scale)

Gaussian process covariance with exponential kernel in multiple dimensions with a length scale for each dimension.

Available since 2.20

matrix gp_exponential_cov(vectors x1, vectors x2, real sigma, real length_scale)

Gaussian process cross-covariance of x1 and x2 with exponential kernel in multiple dimensions.

Available since 2.20

matrix gp_exponential_cov(vectors x1, vectors x2, real sigma, array[] real length_scale)

Gaussian process cross-covariance of x1 and x2 with exponential kernel in multiple dimensions with a length scale for each dimension.

Available since 2.20

### Matern 3/2 kernel

With magnitude $$\sigma$$ and length scale $$l$$, the Matern 3/2 kernel is:

$k(\mathbf{x}_i, \mathbf{x}_j) = \sigma^2 \left( 1 + \frac{\sqrt{3}|\mathbf{x}_i - \mathbf{x}_j|}{l} \right) \exp \left( -\frac{\sqrt{3}|\mathbf{x}_i - \mathbf{x}_j|}{l} \right)$

matrix gp_matern32_cov(array[] real x, real sigma, real length_scale)

Gaussian process covariance with Matern 3/2 kernel in one dimension.

Available since 2.20

matrix gp_matern32_cov(array[] real x1, array[] real x2, real sigma, real length_scale)

Gaussian process cross-covariance of x1 and x2 with Matern 3/2 kernel in one dimension.

Available since 2.20

matrix gp_matern32_cov(vectors x, real sigma, real length_scale)

Gaussian process covariance with Matern 3/2 kernel in multiple dimensions.

Available since 2.20

matrix gp_matern32_cov(vectors x, real sigma, array[] real length_scale)

Gaussian process covariance with Matern 3/2 kernel in multiple dimensions with a length scale for each dimension.

Available since 2.20

matrix gp_matern32_cov(vectors x1, vectors x2, real sigma, real length_scale)

Gaussian process cross-covariance of x1 and x2 with Matern 3/2 kernel in multiple dimensions.

Available since 2.20

matrix gp_matern32_cov(vectors x1, vectors x2, real sigma, array[] real length_scale)

Gaussian process cross-covariance of x1 and x2 with Matern 3/2 kernel in multiple dimensions with a length scale for each dimension.

Available since 2.20

### Matern 5/2 kernel

With magnitude $$\sigma$$ and length scale $$l$$, the Matern 5/2 kernel is:

$k(\mathbf{x}_i, \mathbf{x}_j) = \sigma^2 \left( 1 + \frac{\sqrt{5}|\mathbf{x}_i - \mathbf{x}_j|}{l} + \frac{5 |\mathbf{x}_i - \mathbf{x}_j|^2}{3l^2} \right) \exp \left( -\frac{\sqrt{5} |\mathbf{x}_i - \mathbf{x}_j|}{l} \right)$

matrix gp_matern52_cov(array[] real x, real sigma, real length_scale)

Gaussian process covariance with Matern 5/2 kernel in one dimension.

Available since 2.20

matrix gp_matern52_cov(array[] real x1, array[] real x2, real sigma, real length_scale)

Gaussian process cross-covariance of x1 and x2 with Matern 5/2 kernel in one dimension.

Available since 2.20

matrix gp_matern52_cov(vectors x, real sigma, real length_scale)

Gaussian process covariance with Matern 5/2 kernel in multiple dimensions.

Available since 2.20

matrix gp_matern52_cov(vectors x, real sigma, array[] real length_scale)

Gaussian process covariance with Matern 5/2 kernel in multiple dimensions with a length scale for each dimension.

Available since 2.20

matrix gp_matern52_cov(vectors x1, vectors x2, real sigma, real length_scale)

Gaussian process cross-covariance of x1 and x2 with Matern 5/2 kernel in multiple dimensions.

Available since 2.20

matrix gp_matern52_cov(vectors x1, vectors x2, real sigma, array[] real length_scale)

Gaussian process cross-covariance of x1 and x2 with Matern 5/2 kernel in multiple dimensions with a length scale for each dimension.

Available since 2.20

### Periodic kernel

With magnitude $$\sigma$$, length scale $$l$$, and period $$p$$, the periodic kernel is:

$k(\mathbf{x}_i, \mathbf{x}_j) = \sigma^2 \exp \left(-\frac{2 \sin^2 \left( \pi \frac{|\mathbf{x}_i - \mathbf{x}_j|}{p} \right) }{l^2} \right)$

matrix gp_periodic_cov(array[] real x, real sigma, real length_scale, real period)

Gaussian process covariance with periodic kernel in one dimension.

Available since 2.20

matrix gp_periodic_cov(array[] real x1, array[] real x2, real sigma, real length_scale, real period)

Gaussian process cross-covariance of x1 and x2 with periodic kernel in one dimension.

Available since 2.20

matrix gp_periodic_cov(vectors x, real sigma, real length_scale, real period)

Gaussian process covariance with periodic kernel in multiple dimensions.

Available since 2.20

matrix gp_periodic_cov(vectors x1, vectors x2, real sigma, real length_scale, real period)

Gaussian process cross-covariance of x1 and x2 with periodic kernel in multiple dimensions with a length scale for each dimension.

Available since 2.20

## Linear algebra functions and solvers

### Matrix division operators and functions

In general, it is much more efficient and also more arithmetically stable to use matrix division than to multiply by an inverse. There are specialized forms for lower triangular matrices and for symmetric, positive-definite matrices.

#### Matrix division operators

row_vector operator/(row_vector b, matrix A)
The right division of b by A; equivalently b * inverse(A)

Available since 2.0

matrix operator/(matrix B, matrix A)
The right division of B by A; equivalently B * inverse(A)

Available since 2.5

vector operator\(matrix A, vector b)
The left division of A by b; equivalently inverse(A) * b

Available since 2.18

matrix operator\(matrix A, matrix B)
The left division of A by B; equivalently inverse(A) * B

Available since 2.18

#### Lower-triangular matrix division functions

There are four division functions which use lower triangular views of a matrix. The lower triangular view of a matrix $$\text{tri}(A)$$ is used in the definitions and defined by $\begin{equation*} \text{tri}(A)[m,n] = \left\{ \begin{array}{ll} A[m,n] & \text{if } m \geq n, \text{ and} \\[4pt] 0 & \text{otherwise}. \end{array} \right. \end{equation*}$ When a lower triangular view of a matrix is used, the elements above the diagonal are ignored.

vector mdivide_left_tri_low(matrix A, vector b)
The left division of b by a lower-triangular view of A; algebraically equivalent to the less efficient and stable form inverse(tri(A)) * b, where tri(A) is the lower-triangular portion of A with the above-diagonal entries set to zero.

Available since 2.12

matrix mdivide_left_tri_low(matrix A, matrix B)
The left division of B by a triangular view of A; algebraically equivalent to the less efficient and stable form inverse(tri(A)) * B, where tri(A) is the lower-triangular portion of A with the above-diagonal entries set to zero.

Available since 2.5

row_vector mdivide_right_tri_low(row_vector b, matrix A)
The right division of b by a triangular view of A; algebraically equivalent to the less efficient and stable form b * inverse(tri(A)), where tri(A) is the lower-triangular portion of A with the above-diagonal entries set to zero.

Available since 2.12

matrix mdivide_right_tri_low(matrix B, matrix A)
The right division of B by a triangular view of A; algebraically equivalent to the less efficient and stable form B * inverse(tri(A)), where tri(A) is the lower-triangular portion of A with the above-diagonal entries set to zero.

Available since 2.5

### Symmetric positive-definite matrix division functions

There are four division functions which are specialized for efficiency and stability for symmetric positive-definite matrix dividends. If the matrix dividend argument is not symmetric and positive definite, these will reject and print warnings.

matrix mdivide_left_spd(matrix A, vector b)
The left division of b by the symmetric, positive-definite matrix A; algebraically equivalent to the less efficient and stable form inverse(A) * b.

Available since 2.12

vector mdivide_left_spd(matrix A, matrix B)
The left division of B by the symmetric, positive-definite matrix A; algebraically equivalent to the less efficient and stable form inverse(A) * B.

Available since 2.12

row_vector mdivide_right_spd(row_vector b, matrix A)
The right division of b by the symmetric, positive-definite matrix A; algebraically equivalent to the less efficient and stable form b *inverse(A).

Available since 2.12

matrix mdivide_right_spd(matrix B, matrix A)
The right division of B by the symmetric, positive-definite matrix A; algebraically equivalent to the less efficient and stable form B * inverse(A).

Available since 2.12

### Matrix exponential

The exponential of the matrix $$A$$ is formally defined by the convergent power series: $\begin{equation*} e^A = \sum_{n=0}^{\infty} \dfrac{A^n}{n!} \end{equation*}$

matrix matrix_exp(matrix A)
The matrix exponential of A

Available since 2.13

matrix matrix_exp_multiply(matrix A, matrix B)
The multiplication of matrix exponential of A and matrix B; algebraically equivalent to the less efficient form matrix_exp(A) * B.

Available since 2.18

matrix scale_matrix_exp_multiply(real t, matrix A, matrix B)
The multiplication of matrix exponential of tA and matrix B; algebraically equivalent to the less efficient form matrix_exp(t * A) * B.

Available since 2.18

### Matrix power

Returns the nth power of the specific matrix: $\begin{equation*} M^n = M_1 * ... * M_n \end{equation*}$

matrix matrix_power(matrix A, int B)
Matrix A raised to the power B.

Available since 2.24

### Linear algebra functions

#### Trace

real trace(matrix A)
The trace of A, or 0 if A is empty; A is not required to be diagonal

Available since 2.0

#### Determinants

real determinant(matrix A)
The determinant of A

Available since 2.0

real log_determinant(matrix A)
The log of the absolute value of the determinant of A

Available since 2.0

real log_determinant_spd(matrix A)
The log of the absolute value of the determinant of the symmetric, positive-definite matrix A.

Available since 2.30

#### Inverses

It is almost never a good idea to use matrix inverses directly because they are both inefficient and arithmetically unstable compared to the alternatives. Rather than inverting a matrix m and post-multiplying by a vector or matrix a, as in inverse(m) * a, it is better to code this using matrix division, as in m \ a. The pre-multiplication case is similar, with b * inverse(m) being more efficiently coded as as b / m. There are also useful special cases for triangular and symmetric, positive-definite matrices that use more efficient solvers.

Warning: The function inv(m) is the elementwise inverse function, which returns 1 / m[i, j] for each element.

matrix inverse(matrix A)
Compute the inverse of A

Available since 2.0

matrix inverse_spd(matrix A)
Compute the inverse of A where A is symmetric, positive definite. This version is faster and more arithmetically stable when the input is symmetric and positive definite.

Available since 2.0

matrix chol2inv(matrix L)
Compute the inverse of the matrix whose cholesky factorization is L. That is, for $$A = L L^T$$, return $$A^{-1}$$.

Available since 2.26

#### Generalized Inverse

The generalized inverse $$M^+$$ of a matrix $$M$$ is a matrix that satisfies $$M M^+ M = M$$. For an invertible, square matrix $$M$$, $$M^+$$ is equivalent to $$M^{-1}$$. The dimensions of $$M^+$$ are equivalent to the dimensions of $$M^T$$. The generalized inverse exists for any matrix, so the $$M$$ may be singular or less than full rank.

Even though the generalized inverse exists for any arbitrary matrix, the derivatives of this function only exist on matrices of locally constant rank , meaning, the derivatives do not exist if small perturbations make the matrix change rank. For example, considered the rank of the matrix $$A$$ as a function of $$\epsilon$$:

$A = \left( \begin{array}{cccc} 1 + \epsilon & 2 & 1 \\ 2 & 4 & 2 \end{array} \right)$

When $$\epsilon = 0$$, $$A$$ is rank 1 because the second row is twice the first (and so there is only one linearly independent row). If $$\epsilon \neq 0$$, the rows are no longer linearly dependent, and the matrix is rank 2. This matrix does not have locally constant rank at $$\epsilon = 0$$, and so the derivatives do not exist at zero. Because HMC depends on the derivatives existing, this lack of differentiability creates undefined behavior.

matrix generalized_inverse(matrix A)
The generalized inverse of A

Available since 2.26

#### Eigendecomposition

complex_vector eigenvalues(matrix A)
The complex-valued vector of eigenvalues of the matrix A. The eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix. The eigenvalues are not sorted in any particular order.

Available since 2.30

complex_matrix eigenvectors(matrix A)
The matrix with the complex-valued (column) eigenvectors of the matrix A in the same order as returned by the function eigenvalues

Available since 2.30

tuple(complex_matrix, complex_vector) eigendecompose(matrix A)
Return the matrix of (column) eigenvectors and vector of eigenvalues of the matrix A. This function is equivalent to (eigenvectors(A), eigenvalues(A)) but with a lower computational cost due to the shared work between the two results.

Available since 2.33

vector eigenvalues_sym(matrix A)
The vector of eigenvalues of a symmetric matrix A in ascending order

Available since 2.0

matrix eigenvectors_sym(matrix A)
The matrix with the (column) eigenvectors of symmetric matrix A in the same order as returned by the function eigenvalues_sym

Available since 2.0

tuple(matrix, vector) eigendecompose_sym(matrix A)
Return the matrix of (column) eigenvectors and vector of eigenvalues of the symmetric matrix A. This function is equivalent to (eigenvectors_sym(A), eigenvalues_sym(A)) but with a lower computational cost due to the shared work between the two results.

Available since 2.33

Because multiplying an eigenvector by $$-1$$ results in an eigenvector, eigenvectors returned by a decomposition are only identified up to a sign change. In order to compare the eigenvectors produced by Stan’s eigendecomposition to others, signs may need to be normalized in some way, such as by fixing the sign of a component, or doing comparisons allowing a multiplication by $$-1$$.

The condition number of a symmetric matrix is defined to be the ratio of the largest eigenvalue to the smallest eigenvalue. Large condition numbers lead to difficulty in numerical algorithms such as computing inverses, and thus known as “ill conditioned.” The ratio can even be infinite in the case of singular matrices (i.e., those with eigenvalues of 0).

#### QR decomposition

matrix qr_thin_Q(matrix A)
The orthogonal matrix in the thin QR decomposition of A, which implies that the resulting matrix has the same dimensions as A

Available since 2.18

matrix qr_thin_R(matrix A)
The upper triangular matrix in the thin QR decomposition of A, which implies that the resulting matrix is square with the same number of columns as A

Available since 2.18

tuple(matrix, matrix) qr_thin(matrix A)
Returns both portions of the QR decomposition of A. The first element (“Q”) is the orthonormal matrix in the thin QR decomposition and the second element (“R”) is upper triangular. This function is equivalent to (qr_thin_Q(A), qr_thin_R(A)) but with a lower computational cost due to the shared work between the two results.

Available since 2.33

matrix qr_Q(matrix A)
The orthogonal matrix in the fat QR decomposition of A, which implies that the resulting matrix is square with the same number of rows as A

Available since 2.3

matrix qr_R(matrix A)
The upper trapezoidal matrix in the fat QR decomposition of A, which implies that the resulting matrix will be rectangular with the same dimensions as A

Available since 2.3

tuple(matrix, matrix) qr(matrix A)
Returns both portions of the QR decomposition of A. The first element (“Q”) is the orthonormal matrix in the thin QR decomposition and the second element (“R”) is upper triangular. This function is equivalent to (qr_Q(A), qr_R(A)) but with a lower computational cost due to the shared work between the two results.

Available since 2.33

The thin QR decomposition is always preferable because it will consume much less memory when the input matrix is large than will the fat QR decomposition. Both versions of the decomposition represent the input matrix as $\begin{equation*} A = Q \, R. \end{equation*}$ Multiplying a column of an orthogonal matrix by $$-1$$ still results in an orthogonal matrix, and you can multiply the corresponding row of the upper trapezoidal matrix by $$-1$$ without changing the product. Thus, Stan adopts the normalization that the diagonal elements of the upper trapezoidal matrix are strictly positive and the columns of the orthogonal matrix are reflected if necessary. Also, these QR decomposition algorithms do not utilize pivoting and thus may be numerically unstable on input matrices that have less than full rank.

#### Cholesky decomposition

Every symmetric, positive-definite matrix (such as a correlation or covariance matrix) has a Cholesky decomposition. If $$\Sigma$$ is a symmetric, positive-definite matrix, its Cholesky decomposition is the lower-triangular vector $$L$$ such that $\begin{equation*} \Sigma = L \, L^{\top}. \end{equation*}$

matrix cholesky_decompose(matrix A)
The lower-triangular Cholesky factor of the symmetric positive-definite matrix A

Available since 2.0

#### Singular value decomposition

The matrix A can be decomposed into a diagonal matrix of singular values, D, and matrices of its left and right singular vectors, U and V, $\begin{equation*} A = U D V^T. \end{equation*}$ The matrices of singular vectors here are thin. That is for an $$N$$ by $$P$$ input A, $$M = min(N, P)$$, U is size $$N$$ by $$M$$ and V is size $$P$$ by $$M$$.

vector singular_values(matrix A)
The singular values of A in descending order

Available since 2.0

matrix svd_U(matrix A)
The left-singular vectors of A

Available since 2.26

matrix svd_V(matrix A)
The right-singular vectors of A

Available since 2.26

tuple(matrix, vector, matrix) svd(matrix A)
Returns a tuple containing the left-singular vectors of A, the singular values of A in descending order, and the right-singular values of A. This function is equivalent to (svd_U(A), singular_values(A), svd_V(A)) but with a lower computational cost due to the shared work between the different components.

Available since 2.33

## Sort functions

See the sorting functions section for examples of how the functions work.

vector sort_asc(vector v)
Sort the elements of v in ascending order

Available since 2.0

row_vector sort_asc(row_vector v)
Sort the elements of v in ascending order

Available since 2.0

vector sort_desc(vector v)
Sort the elements of v in descending order

Available since 2.0

row_vector sort_desc(row_vector v)
Sort the elements of v in descending order

Available since 2.0

array[] int sort_indices_asc(vector v)
Return an array of indices between 1 and the size of v, sorted to index v in ascending order.

Available since 2.3

array[] int sort_indices_asc(row_vector v)
Return an array of indices between 1 and the size of v, sorted to index v in ascending order.

Available since 2.3

array[] int sort_indices_desc(vector v)
Return an array of indices between 1 and the size of v, sorted to index v in descending order.

Available since 2.3

array[] int sort_indices_desc(row_vector v)
Return an array of indices between 1 and the size of v, sorted to index v in descending order.

Available since 2.3

int rank(vector v, int s)
Number of components of v less than v[s]

Available since 2.0

int rank(row_vector v, int s)
Number of components of v less than v[s]

Available since 2.0

## Reverse functions

vector reverse(vector v)
Return a new vector containing the elements of the argument in reverse order.

Available since 2.23

row_vector reverse(row_vector v)
Return a new row vector containing the elements of the argument in reverse order.

Available since 2.23
1. The softmax function is so called because in the limit as $$y_n \rightarrow \infty$$ with $$y_m$$ for $$m \neq n$$ held constant, the result tends toward the “one-hot” vector $$\theta$$ with $$\theta_n = 1$$ and $$\theta_m = 0$$ for $$m \neq n$$, thus providing a “soft” version of the maximum function.↩︎