# subs

Symbolic substitution

## Syntax

``snew = subs(s,old,new)``
``snew = subs(s,new)``
``snew = subs(s)``
``sMnew = subs(sM,oldM,newM)``
``sMnew = subs(sM,newM)``
``sMnew = subs(sM)``

## Description

### Substitute Symbolic Scalar Variables and Functions

example

````snew = subs(s,old,new)` returns a copy of `s`, replacing all occurrences of `old` with `new`, and then evaluates `s`. Here, `s` is an expression of symbolic scalar variables or a symbolic function, and `old` specifies the symbolic scalar variables or symbolic function to be substituted. If `old` and `new` are both vectors or cell arrays of the same size, `subs` replaces each element of `old` with the corresponding element of `new`.If `old` is a scalar, and `new` is a vector or matrix, then `subs(s,old,new)` replaces all instances of `old` in `s` with `new`, performing all operations element-wise. All constant terms in `s` are replaced with the constant multiplied by a vector or matrix of all ones. ```

example

````snew = subs(s,new)` returns a copy of `s`, replacing all occurrences of the default symbolic scalar variable in `s` with `new`, and then evaluates `s`. The default variable is defined by `symvar(s,1)`.```

example

````snew = subs(s)` returns a copy of `s`, replacing symbolic scalar variables in `s` with their assigned values in the MATLAB® workspace, and then evaluates `s`. Variables with no assigned values remain as variables.```

### Substitute Symbolic Matrix Variables and Functions

example

````sMnew = subs(sM,oldM,newM)` returns a copy of `sM`, replacing all occurrences of `oldM` with `newM`, and then evaluates `sM`. Here, `sM` is an expression, equation, or condition involving symbolic matrix variables and matrix functions, and `oldM` specifies the symbolic matrix variables and matrix functions to be substituted. The substitution values `newM` must have the same size as `oldM`. (since R2021b)```

example

````sMnew = subs(sM,newM)` returns a copy of `sM`, replacing all occurrences of the default symbolic matrix variable in `sM` with `newM`, and then evaluates `sM`. (since R2021b)```

example

````sMnew = subs(sM)` returns a copy of `sM`, replacing symbolic matrix variables in `sM` with their assigned values in the MATLAB workspace, and then evaluates `sM`. Variables with no assigned values remain as variables. (since R2023b)```

## Examples

collapse all

Replace `a` with `4` in this expression.

```syms a b subs(a + b,a,4)```
`ans = $b+4$`

Replace `a*b` with `5` in this expression.

`subs(a*b^2,a*b,5)`
`ans = $5 b$`

Substitute the default symbolic scalar variable in this expression with `a`. If you do not specify the scalar variable or expression to replace, `subs` uses `symvar` to find the default variable. For `x + y`, the default variable is `x`.

```syms x y a symvar(x + y,1)```
`ans = $x$`

Therefore, `subs` replaces `x` with `a`.

`subs(x + y,a)`
`ans = $a+y$`

When you assign a new value to a symbolic scalar variable, expressions containing the variable are not automatically evaluated. Instead, evaluate expressions by using `subs`.

Define the expression `y = x^2`.

```syms x y = x^2;```

Assign `2` to `x`. The value of `y` is still `x^2` instead of `4`.

```x = 2; y```
`y = ${x}^{2}$`

Evaluate `y` with the new value of `x` by using `subs`.

`subs(y)`
`ans = $4$`

Make multiple substitutions by specifying the old and new values as vectors.

```syms a b subs(cos(a) + sin(b), [a,b], [sym('alpha'),2])```
`ans = $\mathrm{sin}\left(2\right)+\mathrm{cos}\left(\alpha \right)$`

Alternatively, for multiple substitutions, use cell arrays.

`subs(cos(a) + sin(b), {a,b}, {sym('alpha'),2})`
`ans = $\mathrm{sin}\left(2\right)+\mathrm{cos}\left(\alpha \right)$`

Replace the symbolic scalar variable `a` in this expression with the 3-by-3 magic square matrix. Note that the constant `1` expands to the 3-by-3 matrix with all its elements equal to `1`.

```syms a t subs(exp(a*t) + 1, a, -magic(3))```
```ans =  $\left(\begin{array}{ccc}{\mathrm{e}}^{-8 t}+1& {\mathrm{e}}^{-t}+1& {\mathrm{e}}^{-6 t}+1\\ {\mathrm{e}}^{-3 t}+1& {\mathrm{e}}^{-5 t}+1& {\mathrm{e}}^{-7 t}+1\\ {\mathrm{e}}^{-4 t}+1& {\mathrm{e}}^{-9 t}+1& {\mathrm{e}}^{-2 t}+1\end{array}\right)$```

You can also replace an element of a vector, matrix, or array with a nonscalar value. For example, create these 2-by-2 matrices.

`A = sym('A',[2,2])`
```A =  $\left(\begin{array}{cc}{A}_{1,1}& {A}_{1,2}\\ {A}_{2,1}& {A}_{2,2}\end{array}\right)$```
`B = sym('B',[2,2])`
```B =  $\left(\begin{array}{cc}{B}_{1,1}& {B}_{1,2}\\ {B}_{2,1}& {B}_{2,2}\end{array}\right)$```

Replace the first element of the matrix `A` with the matrix `B`. While making this substitution, `subs` expands the 2-by-2 matrix `A` into this 4-by-4 matrix.

`A44 = subs(A, A(1,1), B)`
```A44 =  $\left(\begin{array}{cccc}{B}_{1,1}& {B}_{1,2}& {A}_{1,2}& {A}_{1,2}\\ {B}_{2,1}& {B}_{2,2}& {A}_{1,2}& {A}_{1,2}\\ {A}_{2,1}& {A}_{2,1}& {A}_{2,2}& {A}_{2,2}\\ {A}_{2,1}& {A}_{2,1}& {A}_{2,2}& {A}_{2,2}\end{array}\right)$```

`subs` does not let you replace a nonscalar or matrix with a scalar that shrinks the matrix size.

Create a structure array with symbolic expressions as the field values.

```syms x y z S = struct('f1',x*y,'f2',y + z,'f3',y^2)```
```S = struct with fields: f1: x*y f2: y + z f3: y^2 ```

Replace the symbolic scalar variables `x`, `y`, and `z` with numeric values.

`Sval = subs(S,[x y z],[0.5 1 1.5])`
```Sval = struct with fields: f1: 1/2 f2: 5/2 f3: 1 ```

Replace the symbolic scalar variables `x` and `y` with these 2-by-2 matrices. When you make multiple substitutions involving vectors or matrices, use cell arrays to specify the old and new values.

```syms x y subs(x*y, {x,y}, {[0 1; -1 0], [1 -1; -2 1]})```
```ans =  $\left(\begin{array}{cc}0& -1\\ 2& 0\end{array}\right)$```

Note that because `x` and `y` are scalars, these substitutions are element-wise.

`[0 1; -1 0].*[1 -1; -2 1]`
```ans = 2×2 0 -1 2 0 ```

Eliminate scalar variables from an equation by using the variable's value from another equation. In the second equation, isolate the variable on the left side using `isolate`, and then substitute the right side with the variable in the first equation.

First, declare the equations `eqn1` and `eqn2`.

```syms x y eqn1 = sin(x)+y == x^2 + y^2; eqn2 = y*x == cos(x);```

Isolate `y` in `eqn2` by using `isolate`.

`eqn2 = isolate(eqn2,y)`
```eqn2 =  $y=\frac{\mathrm{cos}\left(x\right)}{x}$```

Eliminate `y` from `eqn1` by substituting the left side of `eqn2` with the right side of `eqn2`.

`eqn1 = subs(eqn1,lhs(eqn2),rhs(eqn2))`
```eqn1 =  $\mathrm{sin}\left(x\right)+\frac{\mathrm{cos}\left(x\right)}{x}=\frac{{\mathrm{cos}\left(x\right)}^{2}}{{x}^{2}}+{x}^{2}$```

Replace `x` with `a` in this symbolic function.

```syms x y a syms f(x,y) f(x,y) = x + y; f = subs(f,x,a)```
`f(x, y) = $a+y$`

`subs` replaces the values in the symbolic function formula, but it does not replace input arguments of the function.

`formula(f)`
`ans = $a+y$`
`argnames(f)`
`ans = $\left(\begin{array}{cc}x& y\end{array}\right)$`

Replace the arguments of a symbolic function explicitly.

```syms x y f(x,y) = x + y; f(a,y) = subs(f,x,a); f```
`f(a, y) = $a+y$`

Suppose you want to verify the solutions of this system of equations.

```syms x y eqs = [x^2 + y^2 == 1, x == y]; S = solve(eqs,[x y]); S.x```
```ans =  $\left(\begin{array}{c}-\frac{\sqrt{2}}{2}\\ \frac{\sqrt{2}}{2}\end{array}\right)$```
`S.y`
```ans =  $\left(\begin{array}{c}-\frac{\sqrt{2}}{2}\\ \frac{\sqrt{2}}{2}\end{array}\right)$```

Verify the solutions by substituting the solutions into the original system.

`isAlways(subs(eqs,S))`
```ans = 2x2 logical array 1 1 1 1 ```

Since R2021b

Define the product of two 2-by-2 matrices. Declare the matrices as symbolic matrix variables with the `symmatrix` data type.

```syms X Y [2 2] matrix sM = X*Y```
`sM = $X Y$`

Replace the matrix variables $X$ and $Y$ with 2-by-2 symbolic matrices. When you make multiple substitutions involving vectors or matrices, use cell arrays to specify the matrix variables to be substituted and their new values. The new values must have the same size as the matrix variables to be substituted.

`S = subs(sM,{X,Y},{[0 sqrt(sym(2)); sqrt(sym(2)) 0], [1 -1; -2 1]})`
```S =  ```

Convert the expression `S` to the `sym` data type to show the result of the substituted matrix multiplication.

`Ssym = symmatrix2sym(S)`
```Ssym =  $\left(\begin{array}{cc}-2 \sqrt{2}& \sqrt{2}\\ \sqrt{2}& -\sqrt{2}\end{array}\right)$```

Since R2021b

Create a matrix of symbolic numbers.

`A = sym([1 4 2; 4 1 2; 2 2 3])`
```A =  $\left(\begin{array}{ccc}1& 4& 2\\ 4& 1& 2\\ 2& 2& 3\end{array}\right)$```

Compute the coefficients of the characteristic polynomial of `A` using the `charpoly` function.

`c = charpoly(A)`
`c = $\left(\begin{array}{cccc}1& -5& -17& 21\end{array}\right)$`

Next, define $X$ as a 3-by-3 symbolic matrix variable. Use the coefficients `c` to create the polynomial $p\left(X\right)={c}_{1}{X}^{3}+{c}_{2}{X}^{2}+{c}_{3}X+{c}_{4}{I}_{3}$, where $X$ is an indeterminate that represents a 3-by-3 matrix.

```syms X [3 3] matrix p = c(1)*X^3 + c(2)*X^2 + c(3)*X + c(4)*X^0```
`p = $21 {\mathrm{I}}_{3}-17 X-5 {X}^{2}+{X}^{3}$`

Substitute $X$ in the polynomial $p\left(X\right)$ with `A` using the `subs` function. According to the Cayley-Hamilton theorem, this substitution results in a 3-by-3 zero matrix because the coefficients `c` are the characteristic polynomial of `A`. Use `symmatrix2sym` to convert the substituted expression to a matrix of symbolic numbers.

`Y = subs(p,A)`
```Y =  ```
`Z = symmatrix2sym(Y)`
```Z =  $\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$```

Since R2022a

Define the function $\mathit{f}\left(\mathbit{A}\right)={\mathbit{A}}^{2}-2\mathbit{A}+{\mathrm{I}}_{2}$, where $\mathbit{A}$ is a 2-by-2 matrix and ${\mathrm{I}}_{2}$ is a 2-by-2 identity matrix. Substitute the variable $\mathbit{A}$ with another expression and evaluate the new function.

Create a 2-by-2 symbolic matrix variable $\mathbit{A}$. Create a symbolic matrix function $\mathit{f}\left(\mathbit{A}\right)$, keeping the existing definition of $\mathbit{A}$ in the workspace. Assign the polynomial expression of $\mathit{f}\left(\mathbit{A}\right)$.

```syms A 2 matrix syms f(A) 2 matrix keepargs f(A) = A*A - 2*A + eye(2)```
`f(A) = ${\mathrm{I}}_{2}-2 A+{A}^{2}$`

Next, create new symbolic matrix variables $\mathbit{B}$ and $\mathbit{C}$. Create a new symbolic matrix function $\mathit{g}\left(\mathbit{B},\mathbit{C}\right)$, keeping the existing definitions of $\mathbit{B}$ and $\mathbit{C}$ in the workspace.

```syms B C 2 matrix syms g(B,C) 2 matrix keepargs```

Substitute the variable $\mathbit{A}$ in $\mathit{f}\left(\mathbit{A}\right)$ with $\mathbit{B}+\mathbit{C}$. Assign the substituted result to the new function $\mathit{g}\left(\mathbit{B},\mathbit{C}\right)$.

`g(B,C) = subs(f,A,B+C)`
`g(B, C) = ${\left(B+C\right)}^{2}+{\mathrm{I}}_{2}-2 B-2 C$`

Evaluate $\mathit{g}\left(\mathbit{B},\mathbit{C}\right)$ for the matrix values $\mathbit{B}=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$ and $\mathbit{C}=\left[\begin{array}{cc}1& -1\\ -2& 1\end{array}\right]$ using `subs`.

`S = subs(g(B,C),{B,C},{[0 1; -1 0],[1 -1; -2 1]})`
```S =  ```

Convert the expression `S` from the `symmatrix` data type to the `sym` data type to show the result of the substituted polynomial.

`Ssym = symmatrix2sym(S)`
```Ssym =  $\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$```

Since R2022b

Define the equation , where $\mathbit{A}$ is a 3-by-3 matrix and $\mathbit{X}$ is a 3-by-1 matrix. Substitute $f\left(\mathbit{X},\mathbit{A}\right)$ with another symbolic expression and $\mathbit{A}$ with symbolic values. Check if the equation is true for these values.

Create two symbolic matrix variables $\mathbit{A}$ and $\mathbit{X}$. Create a symbolic matrix function $f\left(\mathbit{X},\mathbit{A}\right)$, keeping the existing definitions of $\mathbit{A}$ and $\mathbit{X}$ in the workspace. Create the equation.

```syms A [3 3] matrix syms X [3 1] matrix syms f(X,A) [1 1] matrix keepargs eq = diff(diff(f,X),X.') == 2*A```
```eq(X, A) =  ```

Substitute $f\left(\mathbit{X},\mathbit{A}\right)$ with ${\mathbit{X}}^{\mathit{T}}\mathbit{AX}$ and evaluate the second-order differential function in the equation for this expression.

`eq = subs(eq,f,X.'*A*X)`
`eq(X, A) = ${A}^{\mathrm{T}}+A=2 A$`

Substitute $\mathbit{A}$ with the Hilbert matrix of order 3.

`eq = subs(eq,A,hilb(3))`
```eq(X, A) =  ```

Check if the equation is true for these values by using `isAlways`. Because `isAlways` accepts only a symbolic input of type `symfun` or `sym`, convert `eq` from type `symfunmatrix` to type `symfun` before using `isAlways`.

`tf = isAlways(symfunmatrix2symfun(eq))`
```tf = 3x3 logical array 1 1 1 1 1 1 1 1 1 ```

Since R2023b

Define the expression $\mathbit{X}{\mathbit{Y}}^{2}-\mathbit{Y}{\mathbit{X}}^{2}$, where $\mathbit{X}$ and $\mathbit{Y}$ are 3-by-3 matrices. Create the matrices as symbolic matrix variables.

```syms X Y [3 3] matrix C = X*Y^2 - Y*X^2```
`C = $X {Y}^{2}-Y {X}^{2}$`

Assign values to the matrices `X` and `Y`.

```X = [-1 2 pi; 0 1/2 2; 2 1 0]; Y = [3 2 2; -1 2 1; 1 2 -1];```

Evaluate the expression `C` with the assigned values of `X` and `Y` by using `subs`.

`Cnew = subs(C)`
```Cnew =  ```

Convert the result from the `symmatrix` data type to the `double` data type.

`Cnum = double(Cnew)`
```Cnum = 3×3 -42.8496 -13.3584 -13.4336 -0.7168 3.1416 0.0752 -3.2832 29.8584 16.4248 ```

## Input Arguments

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Symbolic input, specified as a symbolic scalar variable, expression, equation, function, array, matrix, or a structure.

Data Types: `sym` | `symfun` | `struct`

Scalar variable to substitute, specified as a symbolic scalar variable, function, expression, array, or a cell array.

Data Types: `sym` | `symfun` | `cell`

New value to substitute with, specified as a number, symbolic number, scalar variable, function, expression, array, structure, or a cell array.

Data Types: `sym` | `symfun` | `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64` | `struct` | `cell`

Symbolic input, specified as a symbolic matrix variable, matrix function, expression, equation, or condition.

Data Types: `symmatrix` | `symfunmatrix`

Matrix variable or function to substitute, specified as a symbolic matrix variable, matrix function, expression, or a cell array.

Data Types: `symmatrix` | `symfunmatrix` | `cell`

New value to substitute with, specified as a number, symbolic number, matrix variable, matrix function, expression, array, or a cell array. `newM` must have the same size as `oldM` or the default symbolic matrix variable in `sM`.

Data Types: `sym` | `symmatrix` | `symfunmatrix` | `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64` | `struct` | `cell`

## Tips

• `subs(s,__)` does not modify `s`. To modify `s`, use `s = subs(s,__)`.

• If `s` is a univariate polynomial and `new` is a numeric matrix, use `polyvalm(sym2poly(s),new)` to evaluate `s` as a matrix. All constant terms are replaced with the constant multiplied by an identity matrix.

• Starting in R2022b, symbolic substitution involving derivatives or the `diff` function follows the input order of the symbolic objects to be substituted. For example, this code performs symbolic substitution involving the `diff` function.

```syms m k x(t) syms x_t x_t_ddot eqSHM = m*diff(x(t),t,2) == -k*x(t); eqSHMnew = subs(eqSHM,[x(t) diff(x(t),t,2)],[x_t x_t_ddot])```
Before R2022b, the code returns this output:
```eqSHMnew = m*x_t_ddot == -k*x_t```

Starting in R2022b, the code returns this output:

```eqSHMnew = 0 == -k*x_t```
The difference in output is due to `subs` now first substituting `x(t)` with `x_t`, resulting in `diff(x_t,t,2)`, which is equal to `0`. To obtain the substitution result as in previous releases, specify the `diff(x(t),t,2)` term first so that it is substituted before the `x(t)` term.
`eqSHMnew = subs(eqSHM,[diff(x(t),t,2) x(t)],[x_t_ddot x_t])`
```eqSHMnew = m*x_t_ddot == -k*x_t```

## Version History

Introduced before R2006a

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