Symbolic 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
Substitute the default variable in this expression with a.
If you do not specify the 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 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 = sin(2) + cos(alpha)
Alternatively, for multiple substitutions, use cell arrays.
subs(cos(a) + sin(b), {a, b}, {sym('alpha'), 2})ans = sin(2) + cos(alpha)
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.
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: [1×1 sym]
f2: [1×1 sym]
f3: [1×1 sym]Replace the symbolic variables x, y, and
z with numeric values.
Sval = subs(S,[x y z],[0.5 1 1.5])
S =
struct with fields:
f1: [1×1 sym]
f2: [1×1 sym]
f3: [1×1 sym]Show the value of each field.
Sval.f1
ans = 1/2
Sval.f2
ans = 5/2
Sval.f3
ans = 1
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 element-wise.
[0 1; -1 0].*[1 -1; -2 1]
ans =
0 -1
2 0Eliminate 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 == cos(x)/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 = sin(x) + cos(x)/x == cos(x)^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) argnames(f)
ans = a + y ans = [ x, y]
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 S.y
ans = -2^(1/2)/2 2^(1/2)/2 ans = -2^(1/2)/2 2^(1/2)/2
Verify the solutions by substituting the solutions into the original system.
isAlways(subs(eqs, S))
ans = 2×2 logical array 1 1 1 1
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.