matlab 的subs,Symbolic substitution

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2023-12-01

subs

Symbolic substitution

Description

subs(s,old,new)

returns a copy of s, replacing all occurrences of

old with new, and then evaluates

subs(s,new) returns a copy of

s, replacing all occurrences of the default variable in

s with new, and then evaluates

s. The default variable is defined by symvar.

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 evaluates 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

Default Substitution Variable

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 =

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 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 =

x^2

Evaluate y with the new value of x by using

subs.

subs(y)

ans =

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)

Alternatively, for multiple substitutions, use cell arrays.

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

ans =

sin(2) + cos(alpha)

Substitute Scalars with Arrays

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]

subs does not let you replace a nonscalar

with a scalar.

Substitute Symbolic 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: [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 =

Substitute Multiple Scalars with Arrays

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 0

Substitutions in Equations

Eliminate 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

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]

Replace the arguments of a symbolic function explicitly.

syms x y

f(x, y) = x + y;

f(a, y) = subs(f, x, a);

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

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

Input Arguments

s — Input

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

structure.

old — Element to substitute

symbolic variable | symbolic expression | symbolic array

Element to substitute, specified as a symbolic variable, expression, or

array.

new — New element

New element to substitute with, specified as a number, symbolic number, variable,

expression, array, or a structure.

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

matlab 的subs,Symbolic substitution

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