diff --git a/src/hyperspectral/hyperspectral/unmixing/rspa.py b/src/hyperspectral/hyperspectral/unmixing/rspa.py
index a33d31f..bf113ac 100644
--- a/src/hyperspectral/hyperspectral/unmixing/rspa.py
+++ b/src/hyperspectral/hyperspectral/unmixing/rspa.py
@@ -3,39 +3,43 @@
 import numpy as np
 
 def compute_alpha(x, y, β):
-    u = x / np.linalg.norm(x, 2)
-    assert np.linalg.norm(x, 2) > np.linalg.norm(y, 2), "x must be larger in norm than y"
-    num = (β * np.linalg.norm(x, 2)**2 - np.linalg.norm(y, 2)**2)
+    nx, ny = np.linalg.norm(x, 2), np.linalg.norm(y, 2)
+    u = x / nx
+    #assert nx > ny, "x must be larger in norm than y"
+    num = (β * nx**2 - ny**2)
     denom = (β * np.dot(u, x)**2 - np.dot(u, y)**2)
-    #print('Numerator:   {}\nDenominator: {}'.format(num, denom))
     return 1 - math.sqrt(1 - num / denom)
 
-def select_key(spectra, d, p, β):
+def select_key(R, d, p, β):
     try:
         from tqdm.autonotebook import trange
     except:
         trange = range
 
-    m = spectra.shape[1]
-    R = spectra
+    m = R.shape[1]
     Y = R
-    Py = np.eye(m)
     k = []
     k1 = []
     e = []
     for i in trange(d, leave=False):
         k.append(np.argmax(np.linalg.norm(Y, 2, axis=1)))
-        u = spectra[k[i]] / np.linalg.norm(spectra[k[i]], 2)
-        R = spectra - np.einsum('n,m->nm',np.matmul(spectra, u), u)
-        e.append(np.sum(np.linalg.norm(R, 2, axis=1)**p))
-        k1.append(np.argmax(np.linalg.norm(R, 2, axis=1)))
-        x = Y[k[i]]
-        y = Y[k1[i]]
-        #x = np.matmul(Py, spectra[k[i]])
-        #y = np.matmul(Py, spectra[k1[i]])
-        α = compute_alpha(x, y, β)
-        Y = Y - α * np.einsum('n,m->nm', np.matmul(Y, u), u)
-        #Py = np.matmul(np.eye(m) - α * np.outer(u, u), Py)
+        u = R[k[i]] / np.linalg.norm(R[k[i]], 2)
+        Ri = R - np.outer(np.matmul(R, u), u)
+        e.append(np.sum(np.linalg.norm(Ri, 2, axis=1)**(p/2)))
+        if i < d - 1:
+            uyi = Y[k[i]] / np.linalg.norm(Y[k[i]], 2)
+            Yi = Y - np.outer(np.matmul(Y, uyi), uyi)
+            normY = np.linalg.norm(Yi, 2, axis=1)
+            k1.append(np.argmax(normY))
+            if normY[k1[i]] < 1e-12:
+                break
+            else:
+                x = Y[k[i]]
+                u = x / np.linalg.norm(x, 2)
+                y = Y[k1[i]]
+                α = compute_alpha(x, y, β)
+                assert α > 0 and α < 1, "α out of range (0,1)"
+                Y = Y - α * np.outer(np.matmul(Y, u), u)
     return k[np.argmin(e)]
 
 def rspa(image, n, d, β=4.0, p=1, tol=1e-8):