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# Backward Substitution

Solve UX = B for X when U is upper triangular matrix

Libraries:
DSP System Toolbox / Math Functions / Matrices and Linear Algebra / Linear System Solvers

## Description

The Backward Substitution block solves the linear system UX = B by simple backward substitution of variables, where:

• U is the upper triangular M-by-M matrix input to the U port.

• B is the M-by-N matrix input to the B port.

• X is the M-by-N output matrix and is the solution of the system of 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 unit-upper 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.

## Ports

### Input

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Specify the input upper triangular M-by-M matrix U through this port. Inputs U and B must have the same number of rows.

If the input is fixed point, it must be a signed integer or a signed fixed point value with a power-of-two slope and zero bias.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `fixed point`
Complex Number Support: Yes

Specify the B matrix in the UX = B equation as an M-by-N matrix or an M-by-1 vector.

When you specify an M-by-1 vector, the block treats the length-M vector input at port B as an M-by-1 matrix. Inputs U and B must have the same number of rows.

If the input is fixed point, it must be a signed integer or a signed fixed point value with a power-of-two slope and zero bias.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `fixed point`
Complex Number Support: Yes

### Output

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The block returns the output as an M-by-N matrix or an M-by-1 vector. The size of the X matrix is same as the size of the B matrix.

When the inputs are fixed-point, the output matrix X is signed only fixed-point.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `fixed point`
Complex Number Support: Yes

## Parameters

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### Main Tab

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 division operation.

Do not select this check box if any elements on the diagonal of U do not have a value of `1`. When you clear the Input U is unit-upper triangular check box, the block always performs the division operation.

Select to optimize simulation speed when the diagonal elements of complex input U are real.

Note

When U is a complex fixed-point signal, you must select either Input U is unit-upper triangular or Diagonal of complex input U is real. When either of these options are selected, the block ignores any imaginary part of the diagonal of U.

#### Dependencies

This parameter is visible only when Input U is unit-upper triangular is not selected.

### Data Types Tab

Note

Floating-point 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.

Specify the rounding mode for fixed-point operations as one of the following:

• `Floor`

• `Ceiling`

• `Convergent`

• `Nearest`

• `Round`

• `Simplest`

• `Zero`

For more details, see rounding mode.

When you select this parameter, the block saturates the result of its fixed-point operation. When you clear this parameter, the block wraps the result of its fixed-point operation. For details on `saturate` and `wrap`, see overflow mode for fixed-point operations.

Product output specifies the data type of the output of a product operation in the Backward Substitution block. For more information on the product output data type, see Fixed-Point Data Types and Multiplication Data Types.

• `Inherit: Inherit via internal rule` — The block inherits the product output data type based on an internal rule. For more information on this rule, see Inherit via Internal Rule.

• `Inherit: Same as first input` — The block specifies the product output data type to be the same as the first input data type.

• `fixdt(1,16,0)` — The block specifies a signed, binary-point, scaled, fixed-point data type with a word length of 16 bits and a fraction length of 0.

Alternatively, you can set the Product output data type by using the Data Type Assistant. To use the assistant, click the button .

For more information on the data type assistant, see Specify Data Types Using Data Type Assistant (Simulink).

Accumulator specifies the data type of the output of an accumulation operation in the Backward Substitution block. For illustrations on how to use the accumulator data type in this block, see Fixed-Point Data Types.

• `Inherit: Inherit via internal rule` — The block inherits the accumulator data type based on an internal rule. For more information on this rule, see Inherit via Internal Rule.

• `Inherit: Same as product output` — The block specifies the accumulator data type to be the same as the product output data type.

• `Inherit: Same as first input` — The block specifies the accumulator data type to be the same as the first input data type.

• `fixdt(1,16,0)` — The block specifies a signed, binary-point scaled, fixed-point data type with a word length of 16 bits and a fraction length of 0.

Alternatively, you can set the Accumulator data type by using the Data Type Assistant. To use the assistant, click the button .

For more information on the data type assistant, see Specify Data Types Using Data Type Assistant (Simulink).

Output specifies the data type of the output of the Backward Substitution block. For more information on the output data type, see Fixed-Point Data Types.

• `Inherit: Same as first input` — The block specifies the output data type to be the same as the first input data type.

• `fixdt(1,16,0)` — The block specifies a signed, binary-point scaled, fixed-point data type with a word length of 16 bits and a fraction length of 0.

Alternatively, you can set the Output data type by using the Data Type Assistant. To use the assistant, click the button.

For more information on the data type assistant, see Specify Data Types Using Data Type Assistant (Simulink).

Specify the minimum value the block can output. Simulink® software uses this minimum value to perform:

• Simulation range checking. See Specify Signal Ranges (Simulink).

• Automatic scaling of fixed-point data types.

Specify the maximum value the block can output. Simulink software uses this maximum value to perform:

• Simulation range checking. See Specify Signal Ranges (Simulink).

• Automatic scaling of fixed-point data types.

Select this parameter to prevent the fixed-point tools from overriding the data types you specify in the block dialog box.

## Block Characteristics

 Data Types `double` | `fixed point` | `integer` | `single` Direct Feedthrough `no` Multidimensional Signals `no` Variable-Size Signals `no` Zero-Crossing Detection `no`

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## Version History

Introduced before R2006a