# fixed.backwardSubstitute

Solve upper-triangular system of equations through backward substitution

## Syntax

``x = fixed.backwardSubstitute(R, C)``
``x = fixed.backwardSubstitute(R, C, outputType)``

## Description

example

````x = fixed.backwardSubstitute(R, C)` performs backward substitution on upper-triangular matrix `R` to compute x = R\C.```
````x = fixed.backwardSubstitute(R, C, outputType)` returns x = R\C, where the data type of output variable, `x`, is specified by `outputType`.```

## Examples

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This example shows how to solve the system of equations ${\left(\mathit{A}}^{\prime }\mathit{A}\right)\mathit{x}=\mathit{B}$ using forward and backward substitution.

Specify the input variables, `A` and `B`.

```rng default; A = gallery('randsvd', [5,3], 1000); b = [1; 1; 1; 1; 1];```

Compute the upper-triangular factor, `R`, of `A`, where $\mathit{A}=\mathit{QR}$.

`R = fixed.qlessQR(A);`

Use forward and backward substitution to compute the value of `X`.

```X = fixed.forwardSubstitute(R,b); X(:) = fixed.backwardSubstitute(R,X)```
```X = 5×1 105 × -0.9088 2.7123 -0.8958 0 0 ```

This solution is equivalent to using the `fixed.qlessQRMatrixSolve` function.

`x = fixed.qlessQRMatrixSolve(A,b) `
```x = 5×1 105 × -0.9088 2.7123 -0.8958 0 0 ```

## Input Arguments

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Upper triangular input, specified as a matrix.

Data Types: `single` | `double` | `fi`
Complex Number Support: Yes

Linear system factor, specified as a matrix.

Data Types: `single` | `double` | `fi`
Complex Number Support: Yes

Output data type, specified as a `numerictype` object or a numeric variable. If `outputType` is specified as a `numerictype` object, the output, `x`, will have the specified data type. If `outputType` is specified as a numeric variable, `x` will have the same data type as the numeric variable.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64` | `logical` | `fi` | `numerictype`

## Output Arguments

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Solution, returned as a matrix satisfying the equation x = R\C.

## Extended Capabilities

Introduced in R2020b

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