# ldivide, .\

Left array division

## Syntax

## Description

`x = `

divides each element of `B`

.\`A`

`A`

by the corresponding element of
`B`

. The sizes of `A`

and
`B`

must be the same or be compatible.

If the sizes of `A`

and `B`

are compatible,
then the two arrays implicitly expand to match each other. For example, if one
of `A`

or `B`

is a scalar, then the scalar is
combined with each element of the other array. Also, vectors with different
orientations (one row vector and one column vector) implicitly expand to form a
matrix.

## Examples

### Divide Two Numeric Arrays

Create two numeric arrays, `A`

and `B`

, and divide the second array, `B`

, into the first, `A`

.

A = ones(2,3); B = [1 2 3; 4 5 6]; x = B.\A

`x = `*2×3*
1.0000 0.5000 0.3333
0.2500 0.2000 0.1667

### Divide Scalar by Numeric Array

Create a scalar, `c`

, and divide it by a numeric array. The result is the same size as the array.

c = 2; D = [1 2 3; 4 5 6]; x = D.\c

`x = `*2×3*
2.0000 1.0000 0.6667
0.5000 0.4000 0.3333

### Divide Row and Column Vectors

Create a 1-by-2 row vector and 3-by-1 column vector and divide them.

a = 1:2; b = (1:3)'; b .\ a

`ans = `*3×2*
1.0000 2.0000
0.5000 1.0000
0.3333 0.6667

The result is a 3-by-2 matrix, where each (i,j) element in the matrix is equal to `b(i) .\ a(j)`

:

$$\mathit{a}=\left[{\mathit{a}}_{1}\text{\hspace{0.17em}}{\mathit{a}}_{2}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathit{b}=\left[\begin{array}{c}{\mathit{b}}_{1}\\ {\mathit{b}}_{2}\\ {\mathit{b}}_{3}\end{array}\right],\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\text{\hspace{0.17em}}\mathit{b}\text{\hspace{0.17em}}.\backslash \text{\hspace{0.17em}}\mathit{a}=\left[\begin{array}{cc}{\mathit{b}}_{1}\text{\hspace{0.17em}}.\backslash \text{\hspace{0.17em}}{\mathit{a}}_{1}& {\mathit{b}}_{1}\text{\hspace{0.17em}}.\backslash \text{\hspace{0.17em}}{\mathit{a}}_{2}\\ {\mathit{b}}_{2}\text{\hspace{0.17em}}.\backslash \text{\hspace{0.17em}}{\mathit{a}}_{1}& {\mathit{b}}_{2}\text{\hspace{0.17em}}.\backslash \text{\hspace{0.17em}}{\mathit{a}}_{2}\\ {\mathit{b}}_{3}\text{\hspace{0.17em}}.\backslash \text{\hspace{0.17em}}{\mathit{a}}_{1}& {\mathit{b}}_{3}\text{\hspace{0.17em}}.\backslash \text{\hspace{0.17em}}{\mathit{a}}_{2}\end{array}\right].$$

### Divide Tables

*Since R2023a*

Create two tables and divide the second table into the first. The row names (if present in both) and variable names must be the same, but do not need to be in the same orders. Rows and variables of the output are in the same orders as the first input.

B = table([1;2],[3;4],VariableNames=["V1","V2"],RowNames=["R1","R2"])

`B=`*2×2 table*
V1 V2
__ __
R1 1 3
R2 2 4

A = table([4;2],[3;1],VariableNames=["V2","V1"],RowNames=["R2","R1"])

`A=`*2×2 table*
V2 V1
__ __
R2 4 3
R1 2 1

x = B .\ A

`x=`*2×2 table*
V1 V2
___ _______
R1 1 0.66667
R2 1.5 1

## Input Arguments

`A`

, `B`

— Operands

scalars | vectors | matrices | multidimensional arrays | tables | timetables

Operands, specified as scalars, vectors, matrices, multidimensional
arrays, tables, or timetables. Inputs `A`

and
`B`

must either be the same size or have sizes that are
compatible (for example, `A`

is an
`M`

-by-`N`

matrix and
`B`

is a scalar or
`1`

-by-`N`

row vector). For more
information, see Compatible Array Sizes for Basic Operations.

If

`A`

or`B`

is an integer data type, then the other input must be the same integer type or be a scalar double. Operands with an integer data type cannot be complex.

Inputs that are tables or timetables must meet the
following conditions:* (since R2023a)*

If an input is a table or timetable, then all its variables must have data types that support the operation.

If only one input is a table or timetable, then the other input must be a numeric or logical array.

If both inputs are tables or timetables, then:

Both inputs must have the same size, or one of them must be a one-row table.

Both inputs must have variables with the same names. However, the variables in each input can be in a different order.

If both inputs are tables and they both have row names, then their row names must be the same. However, the row names in each input can be in a different order.

If both inputs are timetables, then their row times must be the same. However, the row times in each input can be in a different order.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

| `logical`

| `duration`

| `char`

| `table`

| `timetable`

**Complex Number Support: **Yes

## Tips

The element-wise operators

`./`

and`.\`

are related to each other by the equation`A./B = B.\A`

.When dividing integers, use

`idivide`

for more rounding options.MATLAB

^{®}does not support complex integer division.

## Extended Capabilities

### Tall Arrays

Calculate with arrays that have more rows than fit in memory.

The
`ldivide`

function fully supports tall arrays. For more information,
see Tall Arrays.

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

If you use

`ldivide`

with single type and double type operands, the generated code might not produce the same result as MATLAB. See Binary Element-Wise Operations with Single and Double Operands (MATLAB Coder).

### GPU Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

### Thread-Based Environment

Run code in the background using MATLAB® `backgroundPool`

or accelerate code with Parallel Computing Toolbox™ `ThreadPool`

.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

### GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The `ldivide`

function
fully supports GPU arrays. To run the function on a GPU, specify the input data as a `gpuArray`

(Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

### Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

## Version History

**Introduced before R2006a**

### R2023a: Perform operations directly on tables and timetables

The `ldivide`

operator supports operations directly on tables and
timetables without indexing to access their variables. All variables must have data types
that support the operation. For more information, see Direct Calculations on Tables and Timetables.

### R2020b: Implicit expansion change affects `duration`

arrays

Starting in R2020b, `ldivide`

supports implicit expansion when the
arguments are `duration`

arrays. Between R2020a and R2016b,
implicit expansion was supported only for numeric data types.

### R2016b: Implicit expansion change affects arguments for operators

Starting in R2016b with the addition of implicit expansion, some combinations of arguments for basic operations that previously returned errors now produce results. For example, you previously could not add a row and a column vector, but those operands are now valid for addition. In other words, an expression like `[1 2] + [1; 2]`

previously returned a size mismatch error, but now it executes.

If your code uses element-wise operators and relies on the errors that MATLAB previously returned for mismatched sizes, particularly within a `try`

/`catch`

block, then your code might no longer catch those errors.

For more information on the required input sizes for basic array operations, see Compatible Array Sizes for Basic Operations.

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