Documentation 
Solve UX=B for X when U is upper triangular matrix
The Backward Substitution block solves the linear system UX=B by simple backward substitution of variables, where:
U is the upper triangular MbyM matrix input to the U port.
B is the MbyN matrix input to the B port.
The MbyN output matrix X is the solution of the equations. The block does not check the rank of the inputs.
The block uses only the elements in the upper triangle of input U and ignores the lower elements. When you select the Input U is unitupper triangular check box, the block assumes the elements on the diagonal of U are 1s. This is useful when matrix U is the result of another operation, such as an LDL decomposition, that uses the diagonal elements to represent the D matrix.
The block treats a lengthM vector input at port B as an Mby1 matrix.
The following diagram shows the data types used within the Backward Substitution block for fixedpoint signals.
You can set the product output, accumulator, and output data types in the block dialog as discussed in the following section.
The output of the multiplier is in the product output data type when the input is real. When the input is complex, the result of the multiplication is in the accumulator data type. For details on the complex multiplication performed, see Multiplication Data Types.
The Main pane of the Backward Substitution block dialog box appears as follows.
Select this check box only when all elements on the diagonal of U have a value of 1. When you do so, the block optimizes its behavior by skipping an unnecessary divide operation.
Do not select this check box if there are any elements on the diagonal of U that do not have a value of 1. When you clear the Input U is unitupper triangular check box, the block always performs the necessary divide operation.
Select to optimize simulation speed when the diagonal elements of complex input U are real. This parameter is only visible when Input U is unitupper triangular is not selected.
Note: When U is a complex fixedpoint signal, you must select either Input U is unitupper triangular or Diagonal of complex input U is real. In such a case, any imaginary part of the diagonal of U is ignored. 
The Data Types pane of the Backward Substitution block dialog appears as follows.
Note: Floatingpoint inheritance takes precedence over the data type settings defined on this pane. When inputs are floating point, the block ignores these settings, and all internal data types are floating point. 
Select the rounding mode for fixedpoint operations.
Select the overflow mode for fixedpoint operations.
Specify the product output data type. See FixedPoint Data Types and Multiplication Data Types for illustrations depicting the use of the product output data type in this block. You can set it to:
A rule that inherits a data type, for example, Inherit: Inherit via internal rule
An expression that evaluates to a valid data type, for example, fixdt(1,16,0)
Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Product output data type parameter.
See Specify Data Types Using Data Type Assistant for more information.
Specify the accumulator data type. See FixedPoint Data Types for illustrations depicting the use of the accumulator data type in this block. You can set this parameter to:
A rule that inherits a data type, for example, Inherit: Inherit via internal rule
An expression that evaluates to a valid data type, for example, fixdt(1,16,0)
Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Accumulator data type parameter.
See Specify Data Types Using Data Type Assistant for more information.
Specify the output data type. See FixedPoint Data Types for illustrations depicting the use of the output data type in this block. You can set it to:
A rule that inherits a data type, for example, Inherit: Same as first input
An expression that evaluates to a valid data type, for example, fixdt(1,16,0)
Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Output data type parameter.
See Specify Block Output Data Types for more information.
Specify the minimum value that the block should output. The default value is [] (unspecified). Simulink^{®} software uses this value to perform:
Simulation range checking (see Signal Ranges)
Automatic scaling of fixedpoint data types
Specify the maximum value that the block should output. The default value is [] (unspecified). Simulink software uses this value to perform:
Simulation range checking (see Signal Ranges)
Automatic scaling of fixedpoint data types
Select this parameter to prevent the fixedpoint tools from overriding the data types you specify on the block mask.
Port  Supported Data Types 

U 

B 

X 

Cholesky Solver  DSP System Toolbox 
Forward Substitution  DSP System Toolbox 
LDL Solver  DSP System Toolbox 
LevinsonDurbin  DSP System Toolbox 
LU Solver  DSP System Toolbox 
QR Solver  DSP System Toolbox 
See Linear System Solvers for related information.