subs

Symbolic substitution

Description

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 that contains symbolic scalar variables and old specifies the scalar variables to be substituted.

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 values obtained from the calling function and the MATLAB® Workspace, and then evaluates s. Variables with no assigned values remain as variables.

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 that contains symbolic matrix variables and oldM specifies the matrix variables to be substituted. (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)

Examples

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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 since 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 right side of eqn2 with the left side of eqn2 in eqn1.

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 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 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 old and new values.

S = subs(sM,{X,Y},{[0 1; -1 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 =

$\left(\begin{array}{cc}-2& 1\\ -1& 1\end{array}\right)$

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);

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 results in a 3-by-3 zero matrix, since 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 =

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

Input Arguments

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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, expression, array, or a cell array.

Data Types: sym | cell

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

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

Since R2021b

Input, specified as a symbolic matrix variable, or symbolic expression containing matrix variables.

Data Types: symmatrix

Since R2021b

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

Data Types: symmatrix | cell

Since R2021b

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

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

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 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 1s.

• 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.

Introduced before R2006a

Symbolic Math Toolbox Documentation

Mathematical Modeling with Symbolic Math Toolbox

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