From 1a9a60563a2c0fe11feb1088d989a692fc84cf82 Mon Sep 17 00:00:00 2001
From: utensil <utensilcandel@gmail.com>
Date: Fri, 10 May 2024 18:28:56 +0800
Subject: [PATCH] Fix #511

---
 examples/ipython/issue-511.ipynb | 145 ++++++++++++++++++++++++++++++-
 galgebra/mv.py                   |   3 +
 2 files changed, 146 insertions(+), 2 deletions(-)

diff --git a/examples/ipython/issue-511.ipynb b/examples/ipython/issue-511.ipynb
index 5d0ef59f..5902a167 100644
--- a/examples/ipython/issue-511.ipynb
+++ b/examples/ipython/issue-511.ipynb
@@ -9,7 +9,7 @@
     {
      "data": {
       "text/plain": [
-       "(1, e_0)"
+       "(e_0, e_0)"
       ]
      },
      "execution_count": 1,
@@ -28,9 +28,150 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 2,
    "id": "ea2d702a-dd04-4f5c-8794-1efefc8e7ee9",
    "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\frac{1}{8} \\mathbf{e}_{0}$"
+      ],
+      "text/plain": [
+       "e_0/8"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(2*ex)**-3"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "09faf76e-f43c-4023-be91-3f2460de0456",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\frac{1}{65536}$"
+      ],
+      "text/plain": [
+       "1/65536"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(2*ex)**-16"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "91e20a05-ba7d-48cd-8916-d7460fb99e87",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(2*e_0/29 + 3*e_1/29 + 4*e_2/29, 2*e_0/841 + 3*e_1/841 + 4*e_2/841)"
+      ]
+     },
+     "execution_count": 4,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "((2*ex + 3*ey + 4*ez).inv(), (2*ex + 3*ey + 4*ez)**-3)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "bdf6ce81-ab58-4e3b-aa72-6cd320e6062d",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\frac{1}{841}$"
+      ],
+      "text/plain": [
+       "1/841"
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(2*ex + 3*ey + 4*ez).inv() * (2*ex + 3*ey + 4*ez)**-3"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "2a557448-ac80-4b1c-87a0-0d25d731953c",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle 1$"
+      ],
+      "text/plain": [
+       "1"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "((2*ex + 3*ey + 4*ez)**3) * (2*ex + 3*ey + 4*ez)**-3"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "fd88f171-7f74-4112-9123-e986d8d92d76",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle 1$"
+      ],
+      "text/plain": [
+       "1"
+      ]
+     },
+     "execution_count": 7,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(2*ex + 3*ey + 4*ez)**-3 * ((2*ex + 3*ey + 4*ez)**3)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "47ca185d-a790-4ba9-9c45-265572e6675d",
+   "metadata": {},
    "outputs": [],
    "source": []
   }
diff --git a/galgebra/mv.py b/galgebra/mv.py
index 32c2dd2d..7a652123 100644
--- a/galgebra/mv.py
+++ b/galgebra/mv.py
@@ -815,6 +815,9 @@ def __ror__(self, A):  # dot (|) product
     def __pow__(self, n):  # Integer power operator
         if not isinstance(n, int):
             raise ValueError('!!!!Multivector power can only be to integer power!!!!')
+        
+        if n < 0:
+            return (self**(-n)).inv()
 
         result = S.One
         for x in range(n):