Add project 1 with the corresponding source code and report.

2025-08-31 14:03:13 +02:00
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+
+@inreference{noauthor_tridiagonal_2025,
+	title = {Tridiagonal matrix algorithm},
+	rights = {Creative Commons Attribution-{ShareAlike} License},
+	url = {https://en.wikipedia.org/w/index.php?title=Tridiagonal_matrix_algorithm&oldid=1306920126},
+	abstract = {In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system for n unknowns may be written as
+
+  
+    
+      
+        
+          a
+          
+            i
+          
+        
+        
+          x
+          
+            i
+            −
+            1
+          
+        
+        +
+        
+          b
+          
+            i
+          
+        
+        
+          x
+          
+            i
+          
+        
+        +
+        
+          c
+          
+            i
+          
+        
+        
+          x
+          
+            i
+            +
+            1
+          
+        
+        =
+        
+          d
+          
+            i
+          
+        
+        ,
+      
+    
+    \{{\textbackslash}displaystyle a\_\{i\}x\_\{i-1\}+b\_\{i\}x\_\{i\}+c\_\{i\}x\_\{i+1\}=d\_\{i\},\}
+  
+
+where 
+  
+    
+      
+        
+          a
+          
+            1
+          
+        
+        =
+        0
+      
+    
+    \{{\textbackslash}displaystyle a\_\{1\}=0\}
+  
+ and 
+  
+    
+      
+        
+          c
+          
+            n
+          
+        
+        =
+        0
+      
+    
+    \{{\textbackslash}displaystyle c\_\{n\}=0\}
+  
+.
+
+  
+    
+      
+        
+          
+            [
+            
+              
+                
+                  
+                    b
+                    
+                      1
+                    
+                  
+                
+                
+                  
+                    c
+                    
+                      1
+                    
+                  
+                
+                
+                
+                
+                  0
+                
+              
+              
+                
+                  
+                    a
+                    
+                      2
+                    
+                  
+                
+                
+                  
+                    b
+                    
+                      2
+                    
+                  
+                
+                
+                  
+                    c
+                    
+                      2
+                    
+                  
+                
+                
+                
+              
+              
+                
+                
+                  
+                    a
+                    
+                      3
+                    
+                  
+                
+                
+                  
+                    b
+                    
+                      3
+                    
+                  
+                
+                
+                  ⋱
+                
+                
+              
+              
+                
+                
+                
+                  ⋱
+                
+                
+                  ⋱
+                
+                
+                  
+                    c
+                    
+                      n
+                      −
+                      1
+                    
+                  
+                
+              
+              
+                
+                  0
+                
+                
+                
+                
+                  
+                    a
+                    
+                      n
+                    
+                  
+                
+                
+                  
+                    b
+                    
+                      n
+                    
+                  
+                
+              
+            
+            ]
+          
+        
+        
+          
+            [
+            
+              
+                
+                  
+                    x
+                    
+                      1
+                    
+                  
+                
+              
+              
+                
+                  
+                    x
+                    
+                      2
+                    
+                  
+                
+              
+              
+                
+                  
+                    x
+                    
+                      3
+                    
+                  
+                
+              
+              
+                
+                  ⋮
+                
+              
+              
+                
+                  
+                    x
+                    
+                      n
+                    
+                  
+                
+              
+            
+            ]
+          
+        
+        =
+        
+          
+            [
+            
+              
+                
+                  
+                    d
+                    
+                      1
+                    
+                  
+                
+              
+              
+                
+                  
+                    d
+                    
+                      2
+                    
+                  
+                
+              
+              
+                
+                  
+                    d
+                    
+                      3
+                    
+                  
+                
+              
+              
+                
+                  ⋮
+                
+              
+              
+                
+                  
+                    d
+                    
+                      n
+                    
+                  
+                
+              
+            
+            ]
+          
+        
+        .
+      
+    
+    \{{\textbackslash}displaystyle \{{\textbackslash}begin\{bmatrix\}b\_\{1\}\&c\_\{1\}\&\&\&0{\textbackslash}{\textbackslash}a\_\{2\}\&b\_\{2\}\&c\_\{2\}\&\&{\textbackslash}{\textbackslash}\&a\_\{3\}\&b\_\{3\}\&{\textbackslash}ddots \&{\textbackslash}{\textbackslash}\&\&{\textbackslash}ddots \&{\textbackslash}ddots \&c\_\{n-1\}{\textbackslash}{\textbackslash}0\&\&\&a\_\{n\}\&b\_\{n\}{\textbackslash}end\{bmatrix\}\}\{{\textbackslash}begin\{bmatrix\}x\_\{1\}{\textbackslash}{\textbackslash}x\_\{2\}{\textbackslash}{\textbackslash}x\_\{3\}{\textbackslash}{\textbackslash}{\textbackslash}vdots {\textbackslash}{\textbackslash}x\_\{n\}{\textbackslash}end\{bmatrix\}\}=\{{\textbackslash}begin\{bmatrix\}d\_\{1\}{\textbackslash}{\textbackslash}d\_\{2\}{\textbackslash}{\textbackslash}d\_\{3\}{\textbackslash}{\textbackslash}{\textbackslash}vdots {\textbackslash}{\textbackslash}d\_\{n\}{\textbackslash}end\{bmatrix\}\}.\}
+  
+
+For such systems, the solution can be obtained in 
+  
+    
+      
+        O
+        (
+        n
+        )
+      
+    
+    \{{\textbackslash}displaystyle O(n)\}
+  
+ operations instead of 
+  
+    
+      
+        O
+        (
+        
+          n
+          
+            3
+          
+        
+        )
+      
+    
+    \{{\textbackslash}displaystyle O(n{\textasciicircum}\{3\})\}
+  
+ required by Gaussian elimination. A first sweep eliminates the 
+  
+    
+      
+        
+          a
+          
+            i
+          
+        
+      
+    
+    \{{\textbackslash}displaystyle a\_\{i\}\}
+  
+'s, and then an (abbreviated) backward substitution produces the solution. Examples of such matrices commonly arise from the discretization of 1D Poisson equation and natural cubic spline interpolation.
+Thomas' algorithm is not stable in general, but is so in several special cases, such as when the matrix is diagonally dominant (either by rows or columns) or symmetric positive definite; for a more precise characterization of stability of Thomas' algorithm, see Higham Theorem 9.12. If stability is required in the general case, Gaussian elimination with partial pivoting ({GEPP}) is recommended instead.},
+	booktitle = {Wikipedia},
+	urldate = {2025-08-28},
+	date = {2025-08-20},
+	langid = {english},
+	note = {Page Version {ID}: 1306920126},
+	file = {Snapshot:/home/lars/Zotero/storage/LJCAC2BT/index.html:text/html},
+}