Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

Create symbolic variables, expressions, functions, matrices

**The syntaxes sym(A,set) and sym(A,'clear') for
a symbolic object A in the MATLAB**

**Support of character vectors that are not valid
variable names and do not define a number will be removed in a future
release. To create symbolic expressions, first create symbolic variables,
and then use operations on them. For example, use syms x;
x + 1 instead of sym('x + 1'), exp(sym(pi)) instead
of sym('exp(pi)'), and syms f(var1,...varN) instead
of f(var1,...varN) = sym('f(var1,...varN)').**

creates an `A`

= sym('`a`

', ```
[n1
... nM]
```

)`n1`

-by-`...`

-by-`nM`

symbolic
array filled with automatically generated elements. For example, ```
A
= sym('a',[1 3])
```

creates the row vector ```
A = [a1
a2 a3]
```

. The auto-generated elements do not appear in the MATLAB workspace.
For arrays, these elements have the prefix `a`

followed
by the element's index using `_`

as a delimiter,
such as `a1_3_2`

.

`sym(___, `

creates
a symbolic variable or array and sets the assumption that the variable
or all array elements belong to a `set`

)`set`

. Here, `set`

can
be `'real'`

, `'positive'`

, `'integer'`

,
or `'rational'`

.

Create the symbolic variables `x`

and `y`

.

x = sym('x'); y = sym('y');

Create a 1-by-4 symbolic vector `a`

with
auto-generated elements `a1`

, ..., `a4`

.

a = sym('a', [1 4])

a = [ a1, a2, a3, a4]

Format the names of elements of `a`

by using
a format character vector as the first argument. `sym`

replaces `%d`

in
the format character vector with the index of the element to generate
the element names.

a = sym('x_%d', [1 4])

a = [ x_1, x_2, x_3, x_4]

This syntax does not create symbolic variables `x_1`

,
..., `x_4`

in the MATLAB workspace. Access elements
of `a`

using standard indexing methods.

a(1) a(2:3)

ans = x_1 ans = [ x_2, x_3]

Create a 3-by-4 symbolic matrix with automatically
generated elements. The elements are of the form `ai_j`

,
which generates the elements `A1_1`

, ..., `A3_4`

.

A = sym('A',[3 4])

A = [ A1_1, A1_2, A1_3, A1_4] [ A2_1, A2_2, A2_3, A2_4] [ A3_1, A3_2, A3_3, A3_4]

Create a 4-by-4 matrix with the element names `x_1_1`

,
..., `x_4_4`

by using a format character vector as
the first argument. `sym`

replaces `%d`

in
the format character vector with the index of the element to generate
the element names.

B = sym('x_%d_%d',4)

B = [ x_1_1, x_1_2, x_1_3, x_1_4] [ x_2_1, x_2_2, x_2_3, x_2_4] [ x_3_1, x_3_2, x_3_3, x_3_4] [ x_4_1, x_4_2, x_4_3, x_4_4]

This syntax does not create symbolic variables `A1_1`

,
..., `A3_4`

, `x_1_1`

, ..., `x_4_4`

in
the MATLAB workspace. To access an element of a matrix, use parentheses.

A(2,3) B(4,2)

ans = A2_3 ans = x_4_2

Create a 2-by-2-by-2 symbolic array with automatically
generated elements `A1_1_1`

, ..., `A2_2_2`

.

A = sym('a',[2 2 2])

A(:,:,1) = [ a1_1_1, a1_2_1] [ a2_1_1, a2_2_1] A(:,:,2) = [ a1_1_2, a1_2_2] [ a2_1_2, a2_2_2]

Convert numeric values to symbolic numbers
or expressions. Use `sym`

on subexpressions instead
of the entire expression for better accuracy. Using `sym`

on
entire expressions is inaccurate because MATLAB first converts
the expression to a floating-point number, which loses accuracy. `sym`

cannot
always recover this lost accuracy.

inaccurate1 = sym(1/1234567) accurate1 = 1/sym(1234567) inaccurate2 = sym(sqrt(1234567)) accurate2 = sqrt(sym(1234567)) inaccurate3 = sym(exp(pi)) accurate3 = exp(sym(pi))

inaccurate1 = 7650239286923505/9444732965739290427392 accurate1 = 1/1234567 inaccurate2 = 4886716562018589/4398046511104 accurate2 = 1234567^(1/2) inaccurate3 = 6513525919879993/281474976710656 accurate3 = exp(pi)

When creating symbolic numbers with 15 or more digits, use quotation marks to accurately represent the numbers.

inaccurateNum = sym(11111111111111111111) accurateNum = sym('11111111111111111111')

inaccurateNum = 11111111111111110656 accurateNum = 11111111111111111111

When you use quotation marks to create symbolic complex numbers,
specify the imaginary part of a number as `1i`

, `2i`

,
and so on.

sym('1234567 + 1i')

ans = 1234567 + 1i

Create a symbolic expression and a symbolic matrix from anonymous functions associated with MATLAB handles.

h_expr = @(x)(sin(x) + cos(x)); sym_expr = sym(h_expr)

sym_expr = cos(x) + sin(x)

h_matrix = @(x)(x*pascal(3)); sym_matrix = sym(h_matrix)

sym_matrix = [ x, x, x] [ x, 2*x, 3*x] [ x, 3*x, 6*x]

Create the symbolic variables `x`

, `y`

, `z`

,
and `t`

simultaneously assuming that `x`

is
real, `y`

is positive, `z`

integer,
and `t`

is rational.

x = sym('x','real'); y = sym('y','positive'); z = sym('z','integer'); t = sym('t','rational');

Check the assumptions on `x`

, `y`

,
and `z`

using `assumptions`

.

assumptions

ans = [ in(t, 'rational'), in(x, 'real'), 0 < y, in(z, 'integer')]

For further computations, clear the assumptions using `assume`

.

assume([x y z t],'clear') assumptions

ans = Empty sym: 1-by-0

Create a symbolic matrix and set assumptions on each element of that matrix.

A = sym('A%d%d',[2 2],'positive')

A = [ A11, A12] [ A21, A22]

Solve an equation involving the first element of `A`

. MATLAB assumes
that this element is positive.

solve(A(1, 1)^2 - 1, A(1, 1))

ans = 1

Check the assumptions set on the elements of `A`

by
using `assumptions`

.

assumptions(A)

ans = [ 0 < A21, 0 < A11, 0 < A22, 0 < A12]

Clear all previously set assumptions on elements of a symbolic
matrix by using `assume`

.

assume(A,'clear'); assumptions(A)

ans = Empty sym: 1-by-0

Solve the same equation again.

solve(A(1, 1)^2 - 1, A(1, 1))

ans = -1 1

Convert `pi`

to a symbolic
value.

Choose the conversion technique by specifying the optional second
argument, which can be `'r'`

, `'f'`

, `'d'`

,
or `'e'`

. The default is `'r'`

.
See the Input Arguments section for the details about conversion techniques.

r = sym(pi) f = sym(pi,'f') d = sym(pi,'d') e = sym(pi,'e')

r = pi f = 884279719003555/281474976710656 d = 3.1415926535897931159979634685442 e = pi - (198*eps)/359

To create several symbolic variables in one function call, use `syms`

.

`assume`

| `assumeAlso`

| `assumptions`

| `clear`

| ```
clear
all
```

| `double`

| `eps`

| `reset`

| `symfun`

| `syms`

| `symvar`

Was this topic helpful?