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CountRoots

✖
CountRoots[f,x]

gives the number of real roots of the univariate function f in x.

✖
CountRoots[f,{x,a,b}]

gives the number of roots between a and b.

Details

  • CountRoots counts roots with multiplicities.
  • The function f can be polynomial, holomorphic or meromorphic etc.
  • Roots that lie exactly at xa or xb are counted.
  • The limits a and b can be complex, in which case the range is taken to be the closed rectangle Re[a]≤Re[x]≤Re[b]∧Im[a]≤Im[x]≤Im[b].

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Count the number of polynomial roots between 0 and 10:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-urwrv
Out[1]=1

Count roots of a polynomial in a closed rectangle:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-f0tuao
Out[1]=1

Count roots of a real elementary function in a real interval:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-lj00ta
Out[1]=1

Count roots of a holomorphic function in a closed rectangle:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-hj1i1g
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-py4ed6
Out[2]=2

Scope  (20)Survey of the scope of standard use cases

Basic Uses  (8)

Find the number of the real roots:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-c5pbvt
Out[1]=1

Count roots in a real interval:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-ezfwzq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-k7tj8d
Out[2]=2

Count roots in a closed rectangle:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-7zfgj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-be1mdc
Out[2]=2

Count roots in a vertical line segment:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-n07feg
Out[1]=1

Count roots in a horizontal line segment:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-jnyd2r
Out[1]=1

Multiple roots are counted with their multiplicities:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-fg12d2
Out[1]=1

For a root of multiplicity , all the derivatives for also vanish:

In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-7s0wh
Out[2]=2

Roots at the endpoints of the interval are included:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-bbzws8
Out[1]=1

Roots on the boundary of the rectangle are included:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-bc6ceg
Out[1]=1

Real Elementary Functions  (6)

Count the real roots of a high-degree polynomial:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-cn1327
Out[1]=1

Find the number of non-negative roots of an algebraic function involving high-degree radicals:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-la4q9l
Out[1]=1

Count the non-negative roots of a function involving irrational real powers:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-5w2o6
Out[1]=1

Count the real roots of a real exp-log function:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-d677m
Out[1]=1

Count the real roots of a tame real elementary function:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-bldk5k
Out[1]=1

This shows the plot of the function:

In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-cp84ux
Out[2]=2

Count roots of a real elementary function in a bounded interval:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-7ln0q
Out[1]=1

This shows the plot of the function:

In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-l58mq8
Out[2]=2

Holomorphic Functions  (3)

Count roots of a holomorphic elementary function in a closed rectangle:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-33znu
Out[1]=1

Count roots of a holomorphic special function in a closed rectangle:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-c21823
Out[1]=1

Find the number of roots of an analytic function in a bounded real interval:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-7okn
Out[1]=1

The double root at zero is counted with multiplicity:

In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-12img
Out[2]=2

Meromorphic Functions  (3)

Count roots of a meromorphic elementary function in a closed rectangle:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-h45o7n
Out[1]=1

Visualize roots and poles of the function:

In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-gbds66
Out[5]=5

Count roots of a meromorphic special function in a closed rectangle:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-d1cqwz
Out[1]=1

Visualize roots and poles of the function:

In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-rqp9g
Out[2]=2

Find the number of roots of a meromorphic function in a bounded real interval:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-bwchn8
Out[1]=1

Plot the function:

In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-erjprt
Out[2]=2

Applications  (4)Sample problems that can be solved with this function

The number of 17^(th) roots of unity in the closed unit square in the first quadrant:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-cpvagv
Out[2]=2

Roots on the boundary are counted:

In[3]:=3
✖
https://wolfram.com/xid/0rsvljevs-f9rv3z
Out[3]=3

Check that a function has exactly one root in an interval:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-gqxe9q
Out[1]=1

Use FindRoot to approximate the root:

In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-hx6a65
Out[2]=2

Compute a contour integral of logarithmic derivative of a function using the formula , where is the number of roots for a holomorphic function :

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-h6oa9p
Out[1]=1

Compare with the result of numeric integration:

In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-g8j4pr
Out[2]=2

Test stability of equilibria at 0 of linear dynamical systems by counting the roots of the CharacteristicPolynomial[m,x] in the right half-plane:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-sqk4w
In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-bsnrcc

Use a bound for max absolute value of the roots:

In[3]:=3
✖
https://wolfram.com/xid/0rsvljevs-b0j2hx
Out[3]=3

Count the roots in the right half-plane:

In[4]:=4
✖
https://wolfram.com/xid/0rsvljevs-umqcv
Out[4]=4

Since all eigenvalues of m have negative real parts, the equilibrium is asymptotically stable:

In[5]:=5
✖
https://wolfram.com/xid/0rsvljevs-drgcjd
Out[5]=5

Try a different matrix:

In[6]:=6
✖
https://wolfram.com/xid/0rsvljevs-bz5p5f

This time there are roots with non-negative real parts:

In[7]:=7
✖
https://wolfram.com/xid/0rsvljevs-jd43si
Out[7]=7

The equilibrium is not asymptotically stable:

In[8]:=8
✖
https://wolfram.com/xid/0rsvljevs-jnsq66
Out[8]=8

Properties & Relations  (5)Properties of the function, and connections to other functions

The number of complex roots of a polynomial is equal to its degree:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-xtqs

This gives a bound on absolute values of roots of a polynomial:

In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-h5yn4e
In[3]:=3
✖
https://wolfram.com/xid/0rsvljevs-bcvpmz
Out[3]=3

The polynomial indeed has 10 roots within the Cauchy bounded region:

In[4]:=4
✖
https://wolfram.com/xid/0rsvljevs-erbu1
Out[4]=4

The number of real roots of a polynomial with nonzero terms is at most :

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-dgov4s
Out[1]=1

This polynomial has the maximal possible number of real roots:

In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-iicn2c
Out[2]=2

Use Reduce to find polynomial roots:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-9o38o

Count roots:

In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-tl6u
Out[2]=2

Find roots:

In[3]:=3
✖
https://wolfram.com/xid/0rsvljevs-ddyrvf
Out[3]=3

Use RootIntervals to find isolating intervals for roots:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-gou3i4

Count the real roots:

In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-bsjwhh
Out[2]=2

Isolate the real roots:

In[3]:=3
✖
https://wolfram.com/xid/0rsvljevs-bcgo0s
Out[3]=3

Use NumberFieldSignature to count the real roots and the pairs of complex roots of a polynomial:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-6wlsk
In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-cufel
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0rsvljevs-itd11g
Out[3]=3

Possible Issues  (1)Common pitfalls and unexpected behavior

Roots at the endpoints of the interval are included:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-m9osga
Out[1]=1

Neat Examples  (1)Surprising or curious use cases

Count the roots of a meromorphic function:

In[1]:=1
✖
https://wolfram.com/xid/0rsvljevs-ksmp4f
Out[1]=1

Plot the roots as spikes:

In[2]:=2
✖
https://wolfram.com/xid/0rsvljevs-l7k3hi
Out[2]=2
Wolfram Research (2007), CountRoots, Wolfram Language function, https://reference.wolfram.com/language/ref/CountRoots.html (updated 2017).
✖
Wolfram Research (2007), CountRoots, Wolfram Language function, https://reference.wolfram.com/language/ref/CountRoots.html (updated 2017).

Text

Wolfram Research (2007), CountRoots, Wolfram Language function, https://reference.wolfram.com/language/ref/CountRoots.html (updated 2017).

✖
Wolfram Research (2007), CountRoots, Wolfram Language function, https://reference.wolfram.com/language/ref/CountRoots.html (updated 2017).

CMS

Wolfram Language. 2007. "CountRoots." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/CountRoots.html.

✖
Wolfram Language. 2007. "CountRoots." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/CountRoots.html.

APA

Wolfram Language. (2007). CountRoots. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CountRoots.html

✖
Wolfram Language. (2007). CountRoots. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CountRoots.html

BibTeX

@misc{reference.wolfram_2025_countroots, author="Wolfram Research", title="{CountRoots}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/CountRoots.html}", note=[Accessed: 09-May-2025 ]}

✖
@misc{reference.wolfram_2025_countroots, author="Wolfram Research", title="{CountRoots}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/CountRoots.html}", note=[Accessed: 09-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_countroots, organization={Wolfram Research}, title={CountRoots}, year={2017}, url={https://reference.wolfram.com/language/ref/CountRoots.html}, note=[Accessed: 09-May-2025 ]}

✖
@online{reference.wolfram_2025_countroots, organization={Wolfram Research}, title={CountRoots}, year={2017}, url={https://reference.wolfram.com/language/ref/CountRoots.html}, note=[Accessed: 09-May-2025 ]}