fixed.backwardSubstitute

Solve upper-triangular system of equations through backward substitution

Since R2020b

Syntax

x = fixed.backwardSubstitute(R, C)

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

Description

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

Solve a System of Equations Using Forward and Backward Substitution

This example shows how to solve the system of equations ${(\mathit{A}}^{\prime }\mathit{A})\mathit{x}=\mathit{B}$ 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

R — Upper-triangular input matrix

matrix

Upper triangular input, specified as a matrix.

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

C — Linear system factor

matrix

Linear system factor, specified as a matrix.

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

outputType — Output data type

numerictype object | numeric variable

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

x — Solution

matrix

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

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Generate code for double-precision, single-precision, and fixed-point data types.

Fixed-Point Conversion

Design and simulate fixed-point systems using Fixed-Point Designer™.

R and C must be signed and use binary-point scaling. Slope-bias representation is not supported for fixed-point data types.

Version History

Introduced in R2020b