diff --git a/chapter13/LDS/kalmanFilter.m b/chapter13/LDS/kalmanFilter.m
index 0005ee6..19a4137 100644
--- a/chapter13/LDS/kalmanFilter.m
+++ b/chapter13/LDS/kalmanFilter.m
@@ -32,11 +32,11 @@
 llh(1) = logGauss(X(:,1),C*mu0,R);
 for i = 2:n
     [mu(:,i), V(:,:,i), llh(i)] = ...
-        forwardStep(X(:,i), mu(:,i-1), V(:,:,i-1), A, G, C, S, I);
+        forwardUpdate(X(:,i), mu(:,i-1), V(:,:,i-1), A, G, C, S, I);
 end
 llh = sum(llh);
 
-function [mu, V, llh] = forwardStep(x, mu, V, A, G, C, S, I)
+function [mu, V, llh] = forwardUpdate(x, mu, V, A, G, C, S, I)
 P = A*V*A'+G;                                               % 13.88
 PC = P*C';                                                      
 R = C*PC+S;
diff --git a/chapter13/LDS/kalmanSmoother.m b/chapter13/LDS/kalmanSmoother.m
index 8254230..f6ed2e9 100644
--- a/chapter13/LDS/kalmanSmoother.m
+++ b/chapter13/LDS/kalmanSmoother.m
@@ -1,4 +1,4 @@
-function [nu, U, Ezz, Ezy, llh] = kalmanSmoother(model, X)
+function [nu, U, llh, Ezz, Ezy] = kalmanSmoother(model, X)
 % Kalman smoother (forward-backward algorithm for linear dynamic system)
 % NOTE: This is the exact implementation of the Kalman smoother algorithm in PRML.
 % However, this algorithm is not practical. It is numerical unstable. 
@@ -26,20 +26,19 @@
 P = zeros(q,q,n); % C_{t+1|t}
 Amu = zeros(q,n); % u_{t+1|t}
 llh = zeros(1,n);
-I = eye(q);
 
 % forward
 PC = P0*C';
 R = C*PC+S;
 K = PC/R;
 mu(:,1) = mu0+K*(X(:,1)-C*mu0);
-V(:,:,1) = (I-K*C)*P0;
+V(:,:,1) = (eye(q)-K*C)*P0;
 P(:,:,1) = P0;  % useless, just make a point
 Amu(:,1) = mu0; % useless, just make a point
 llh(1) = logGauss(X(:,1),C*mu0,R);
 for i = 2:n    
     [mu(:,i), V(:,:,i), Amu(:,i), P(:,:,i), llh(i)] = ...
-        forwardStep(X(:,i), mu(:,i-1), V(:,:,i-1), A, G, C, S, I);
+        forwardUpdate(X(:,i), mu(:,i-1), V(:,:,i-1), A, G, C, S);
 end
 llh = sum(llh);
 % backward
@@ -53,24 +52,25 @@
 Ezz(:,:,n) = U(:,:,n)+nu(:,n)*nu(:,n)';
 for i = n-1:-1:1  
     [nu(:,i), U(:,:,i), Ezz(:,:,i), Ezy(:,:,i)] = ...
-        backwardStep(nu(:,i+1), U(:,:,i+1), mu(:,i), V(:,:,i), Amu(:,i+1), P(:,:,i+1), A);
+        backwardUpdate(nu(:,i+1), U(:,:,i+1), mu(:,i), V(:,:,i), Amu(:,i+1), P(:,:,i+1), A);
 end
 
-function [mu, V, Amu, P, llh] = forwardStep(x, mu0, V0, A, G, C, S, I)
+function [mu1, V1, Amu, P, llh] = forwardUpdate(x, mu0, V0, A, G, C, S)
+k = numel(mu0);
 P = A*V0*A'+G;                                              % 13.88
 PC = P*C';
 R = C*PC+S;
 K = PC/R;                                                   % 13.92
 Amu = A*mu0;
 CAmu = C*Amu;
-mu = Amu+K*(x-CAmu);                                        % 13.89
-V = (I-K*C)*P;                                              % 13.90
+mu1 = Amu+K*(x-CAmu);                                        % 13.89
+V1 = (eye(k)-K*C)*P;                                              % 13.90
 llh = logGauss(x,CAmu,R);                                   % 13.91
 
 
-function [nu, U, Ezz, Ezy] = backwardStep(nu0, U0, mu, V, Amu, P, A)
-J = V*A'/P;                                                 % 13.102
-nu = mu+J*(nu0-Amu);                                        % 13.100
-U = V+J*(U0-P)*J';                                          % 13.101
-Ezy = J*U0+nu0*nu';                                         % 13.106 
-Ezz = U+nu*nu';                                             % 13.107
\ No newline at end of file
+function [nu0, U0, E00, E10] = backwardUpdate(nu1, U1, mu, V, Amu, P, A)
+J = V*A'/P;                                                  % 13.102
+nu0 = mu+J*(nu1-Amu);                                        % 13.100
+U0 = V+J*(U1-P)*J';                                          % 13.101
+E00 = U0+nu0*nu0';                                           % 13.107
+E10 = U1*J'+nu1*nu0';                                        % 13.106