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

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.

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).

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

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)

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)

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

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 = $$5\hspace{0.17em}b$$

Default Substitution Variable

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 = $$a+y$$

Evaluate Expression with New Values

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 = $$4$$

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 = $$\sin \left(2\right)+\cos \left(\alpha \right)$$

Alternatively, for multiple substitutions, use cell arrays.

subs(cos(a) + sin(b), {a,b}, {sym("alpha"),2})

ans = $$\sin \left(2\right)+\cos \left(\alpha \right)$$

Substitute Scalars with Arrays

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 = 
$$\left(\begin{array}{ccc}{\mathrm{e}}^{-8\hspace{0.17em}t}+1& {\mathrm{e}}^{-t}+1& {\mathrm{e}}^{-6\hspace{0.17em}t}+1\\ {\mathrm{e}}^{-3\hspace{0.17em}t}+1& {\mathrm{e}}^{-5\hspace{0.17em}t}+1& {\mathrm{e}}^{-7\hspace{0.17em}t}+1\\ {\mathrm{e}}^{-4\hspace{0.17em}t}+1& {\mathrm{e}}^{-9\hspace{0.17em}t}+1& {\mathrm{e}}^{-2\hspace{0.17em}t}+1\end{array}\right)$$

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.

Substitute Symbolic Scalar Variables in Structure Array

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

Substitute Multiple Scalars with Arrays

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

Substitutions in Equations

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 = 
$$y=\frac{\cos \left(x\right)}{x}$$

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 = 
$$\sin \left(x\right)+\frac{\cos \left(x\right)}{x}=\frac{{\cos \left(x\right)}^2}{x^2}+x^2$$

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 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) = $$a+y$$

Substitute Variables with Corresponding Values from Structure

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

Substitute Symbolic Matrix Variables with Arrays

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 = $$X\hspace{0.17em}Y$$

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 = 
$$\begin{array}{l}{\Sigma }_1\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\Sigma }_1=\left(\begin{array}{cc}-2\hspace{0.17em}\sqrt{2}& \sqrt{2}\\ \sqrt{2}& -\sqrt{2}\end{array}\right)\end{array}$$

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

Ssym = symmatrix2sym(S)

Ssym = 
$$\left(\begin{array}{cc}-2\hspace{0.17em}\sqrt{2}& \sqrt{2}\\ \sqrt{2}& -\sqrt{2}\end{array}\right)$$

Characteristic Polynomial of Matrix

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 $$p(X)=c_1{X}^3+c_2{X}^2+c_{3}X+c_4{I}_3$$, 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 = $$21\hspace{0.17em}{\mathrm{I}}_3-17\hspace{0.17em}X-5\hspace{0.17em}{X}^2+X^3$$

Substitute $$X$$ in the polynomial $$p(X)$$ 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 = 
$$\begin{array}{l}-17\hspace{0.17em}{\Sigma }_1-5\hspace{0.17em}{{\Sigma }_1}^2+{{\Sigma }_1}^3+21\hspace{0.17em}{\mathrm{I}}_3\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\Sigma }_1=\left(\begin{array}{ccc}1& 4& 2\\ 4& 1& 2\\ 2& 2& 3\end{array}\right)\end{array}$$

Z = symmatrix2sym(Y)

Z = 
$$\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$$

Substitute Variables in Symbolic Matrix Function

Since R2022a

Define the function $\mathit{f}\left(\mathit{A}\right)={\mathit{A}}^2-2\mathit{A}+{\mathrm{I}}_2$, where $\mathit{A}$ is a 2-by-2 matrix and ${\mathrm{I}}_2$ is a 2-by-2 identity matrix. Substitute the variable $\mathit{A}$ with another expression and evaluate the new function.

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

syms A 2 matrix
syms f(A) 2 matrix keepargs
f(A) = A*A - 2*A + eye(2)

f(A) = $${\mathrm{I}}_2-2\hspace{0.17em}A+A^2$$

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

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

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

g(B,C) = subs(f,A,B+C)

g(B, C) = $${\left(B+C\right)}^2+{\mathrm{I}}_2-2\hspace{0.17em}B-2\hspace{0.17em}C$$

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

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

S = 
$$\begin{array}{l}-2\hspace{0.17em}{\Sigma }_1-2\hspace{0.17em}{\Sigma }_2+{\left({\Sigma }_1+{\Sigma }_2\right)}^2+{\mathrm{I}}_2\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\Sigma }_1=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\\ \\ \mathrm{ }{\Sigma }_2=\left(\begin{array}{cc}1& -1\\ -2& 1\end{array}\right)\end{array}$$

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 = 
$$\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$$

Substitute Symbolic Matrix Variables and Matrix Functions in Equation

Since R2022b

Define the equation $\frac{\partial }{\partial X^{\mathrm{T}}}\mathrm{ }\frac{\partial }{\partial X}\mathrm{ }f\left(X,A\right)=2\hspace{0.17em}A$, where $\mathit{A}$ is a 3-by-3 matrix and $\mathit{X}$ is a 3-by-1 matrix. Substitute $f(\mathit{X},\mathit{A})$ with another symbolic expression and $\mathit{A}$ with symbolic values. Check if the equation is true for these values.

Create two symbolic matrix variables $\mathit{A}$ and $\mathit{X}$. Create a symbolic matrix function $f(\mathit{X},\mathit{A})$, keeping the existing definitions of $\mathit{A}$ and $\mathit{X}$ 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) = 
$$\frac{\partial }{\partial X^{\mathrm{T}}}\mathrm{ }\frac{\partial }{\partial X}\mathrm{ }f\left(X,A\right)=2\hspace{0.17em}A$$

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

eq = subs(eq,f,X.'*A*X)

eq(X, A) = $$A^{\mathrm{T}}+A=2\hspace{0.17em}A$$

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

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

eq(X, A) = 
$$\begin{array}{l}{\Sigma }_1+{{\Sigma }_1}^{\mathrm{T}}=2\hspace{0.17em}{\Sigma }_1\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\Sigma }_1=\left(\begin{array}{ccc}1& \frac{1}{2}& \frac{1}{3}\\ \frac{1}{2}& \frac{1}{3}& \frac{1}{4}\\ \frac{1}{3}& \frac{1}{4}& \frac{1}{5}\end{array}\right)\end{array}$$

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

Evaluate Expression Involving Symbolic Matrix Variables

Since R2023b

Define the expression $\mathit{X}{\mathit{Y}}^2-\mathit{Y}{\mathit{X}}^2$, where $\mathit{X}$ and $\mathit{Y}$ 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 = $$X\hspace{0.17em}{Y}^2-Y\hspace{0.17em}{X}^2$$

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 = 
$$\begin{array}{l}-{\Sigma }_1\hspace{0.17em}{{\Sigma }_2}^2+{\Sigma }_2\hspace{0.17em}{{\Sigma }_1}^2\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\Sigma }_1=\left(\begin{array}{ccc}3& 2& 2\\ -1& 2& 1\\ 1& 2& -1\end{array}\right)\\ \\ \mathrm{ }{\Sigma }_2=\left(\begin{array}{ccc}-1& 2& \pi \\ 0& \frac{1}{2}& 2\\ 2& 1& 0\end{array}\right)\end{array}$$

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

s — Symbolic input

symbolic scalar variable | symbolic expression | symbolic equation | symbolic function | symbolic array | symbolic matrix | structure

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

Data Types: sym | symfun | struct

old — Scalar variable to substitute

symbolic scalar variable | symbolic function | symbolic expression | symbolic array | cell array

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

Data Types: sym | symfun | cell

new — New value

number | symbolic number | symbolic scalar variable | symbolic function | symbolic expression | symbolic array | structure | cell array

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

sM — Symbolic input

symbolic matrix variable | symbolic matrix function | symbolic expression | symbolic equation | symbolic condition

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

Data Types: symmatrix | symfunmatrix

oldM — Matrix variable or function to substitute

symbolic matrix variable | symbolic matrix function | symbolic expression | cell array

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

Data Types: symmatrix | symfunmatrix | cell

newM — New value

number | symbolic number | symbolic matrix variable | symbolic matrix function | symbolic expression | symbolic array | cell array

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

R2023b: Substitute symbolic matrix variables with their workspace values

You can use the syntax subs(sM) to substitute the symbolic matrix variables and matrix functions in sM with their assigned values in the MATLAB workspace and then evaluate sM. Variables with no assigned values remain as variables. For an example, see Evaluate Expression Involving Symbolic Matrix Variables.

R2022b: Substitute symbolic matrix variables and matrix functions in symbolic equations or conditions

The subs function accepts a symbolic equation or condition of type symmatrix or symfunmatrix as the first input argument. For an example, see Substitute Symbolic Matrix Variables and Matrix Functions in Equation.

R2022a: Substitute variables in symbolic matrix functions

The subs function accepts a symbolic matrix function of type symfunmatrix as the first input argument. For an example, see Substitute Variables in Symbolic Matrix Function.

R2021b: Substitute symbolic matrix variables in symbolic expressions

The subs function accepts a symbolic expression of type symmatrix as the first input argument. For examples, see Substitute Symbolic Matrix Variables with Arrays and Characteristic Polynomial of Matrix.