# Documentation

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# subs

Symbolic substitution

## Syntax

• ``subs(s,old,new)``
example
• ``subs(s,new)``
example
• ``subs(s)``
example

## Description

example

````subs(s,old,new)` returns a copy of `s` replacing all occurrences of `old` with `new`, and then evaluating `s`.```

example

````subs(s,new)` returns a copy of `s` replacing all occurrences of the default variable in `s` with `new`, and then evaluating `s`. The default variable is defined by `symvar`.```

example

````subs(s)` returns a copy of `s` replacing symbolic variables in `s` with their values obtained from the calling function and the MATLAB® workspace, and then evaluating `s`. Variables with no assigned values remain as variables.```

## Examples

### Single Substitution

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```

### Value That Gets Substituted by Default

Substitute the default value in this expression with `a`. If you do not specify which variable or expression that you want 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```

### Single Input

Solve this ordinary differential equation.

```syms a y(t) y = dsolve(diff(y) == -a*y)```
```y = C3*exp(-a*t)```

Now, specify the values of the symbolic parameters `a` and `C2`.

```a = 980; C2 = 3;```

Although the values `a` and `C2` are now in the MATLAB workspace, `y` is not evaluated with the account of these values.

`y`
```y = C3*exp(-a*t)```

To evaluate `y` taking into account the new values of `a` and `C2`, use `subs`.

`subs(y)`
```ans = C3*exp(-980*t)```

### Multiple Substitutions

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 = sin(2) + cos(alpha)```

You also can use cell arrays for that purpose.

`subs(cos(a) + sin(b), {a, b}, {sym('alpha'), 2})`
```ans = sin(2) + cos(alpha)```

### Scalar and Matrix Expansion

Replace 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 = [ exp(-8*t) + 1, exp(-t) + 1, exp(-6*t) + 1] [ exp(-3*t) + 1, exp(-5*t) + 1, exp(-7*t) + 1] [ exp(-4*t) + 1, exp(-9*t) + 1, exp(-2*t) + 1]```

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]) B = sym('B', [2,2])```
```A = [ A1_1, A1_2] [ A2_1, A2_2] B = [ B1_1, B1_2] [ B2_1, B2_2]```

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 = [ B1_1, B1_2, A1_2, A1_2] [ B2_1, B2_2, A1_2, A1_2] [ A2_1, A2_1, A2_2, A2_2] [ A2_1, A2_1, A2_2, A2_2]```

`subs` does not let you replace a nonscalar with a scalar.

### Multiple Scalar Expansion

Replace 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 = [ 0, -1] [ 2, 0]```

Note that these substitutions are elementwise.

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

### Substitutions in Equations

Replace `sin(x + 1)` with `a` in this equation.

```syms x a subs(sin(x + 1) + 1 == x, sin(x + 1), a)```
```ans = a + 1 == x```

### Substitutions in Functions

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 does not replace input arguments of the function.

```formula(f) argnames(f)```
```ans = a + y ans = [ x, y]```

You can 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```

### Original Expression

Assign the expression `x + y` to `s`.

```syms x y s = x + y;```

Replace `y` in this expression with the value `1`. Here, `s` itself does not change.

```subs(s, y, 1); s```
```s = x + y```

To replace the value of `s` with the new expression, assign the result returned by `subs` to `s`.

```s = subs(s, y, 1); s```
```s = x + 1```

### Structure Array

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 S.y```
```ans = -2^(1/2)/2 2^(1/2)/2 ans = -2^(1/2)/2 2^(1/2)/2```

To verify the correctness of the returned solutions, substitute the solutions into the original system.

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

## Input Arguments

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Input specified as a symbolic variable, expression, equation, function, array, vector, or matrix.

Existing element that needs to be replaced specified as a symbolic variable, expression, character vector, array, vector, or matrix.

New element specified as a number, variable, expression, character vectors, array, vector, matrix, or structure array.

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### Tips

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

• If `old` and `new` are both vectors or cell arrays of the same size, `subs` replaces each element of `old` by 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 elementwise. All constant terms in `s` are replaced with the constant times a vector or matrix of all 1s.

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