Modified doc/make.py to run sphinx-build -b linkcheck (#54265)

* Modified doc/make.py to run sphinx-build -b linkcheck

* Update make.py

* Update make.py

* Update make.py

* Update make.py

* Update make.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

* Update doc/make.py

Co-authored-by: Matthew Roeschke <10647082+mroeschke@users.noreply.github.com>

* Update make.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

* Update make.py

* Update doc/make.py

Co-authored-by: Philip Meier <github.pmeier@posteo.de>

* Update doc/make.py

Co-authored-by: Philip Meier <github.pmeier@posteo.de>

* Update make.py

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Matthew Roeschke <10647082+mroeschke@users.noreply.github.com>
Co-authored-by: Philip Meier <github.pmeier@posteo.de>

doc/make.py

@@ -123,14 +123,14 @@ def _sphinx_build(self, kind: str):
 
         Parameters
         ----------
-        kind : {'html', 'latex'}
+        kind : {'html', 'latex', 'linkcheck'}
 
         Examples
         --------
         >>> DocBuilder(num_jobs=4)._sphinx_build('html')
         """
-        if kind not in ("html", "latex"):
-            raise ValueError(f"kind must be html or latex, not {kind}")
+        if kind not in ("html", "latex", "linkcheck"):
+            raise ValueError(f"kind must be html, latex or linkcheck, not {kind}")
 
         cmd = ["sphinx-build", "-b", kind]
         if self.num_jobs:
@@ -288,6 +288,12 @@ def zip_html(self):
         os.chdir(dirname)
         self._run_os("zip", zip_fname, "-r", "-q", *fnames)
 
+    def linkcheck(self):
+        """
+        Check for broken links in the documentation.
+        """
+        return self._sphinx_build("linkcheck")
+
 
 def main():
     cmds = [method for method in dir(DocBuilder) if not method.startswith("_")]