# LinearAlgebraOperationsExtensions.BandTriangularMultiplyInPlace<T>(ILinearAlgebraOperations<T>, MatrixTriangle, TransposeOperation, MatrixDiagonal, Int32, Int32, ReadOnlySpan2D<T>, SpanSlice<T>) Method

Performs one of the matrix-vector operations
x := A*x, or x := A^{T}*x,
where x is an n element vector and A is an n by n unit, or non-unit,
upper or lower triangular band matrix, with ( k + 1 ) diagonals.

## Definition

**Namespace:**Extreme.Collections

**Assembly:**Extreme.Numerics (in Extreme.Numerics.dll) Version: 9.0.0

```
public static void BandTriangularMultiplyInPlace<T>(
this ILinearAlgebraOperations<T> operations,
MatrixTriangle uplo,
TransposeOperation trans,
MatrixDiagonal diag,
int n,
int k,
ReadOnlySpan2D<T> a,
SpanSlice<T> x
)
```

#### Parameters

- operations ILinearAlgebraOperations<T>
- uplo MatrixTriangle
On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' A is an upper triangular matrix. UPLO = 'L' or 'l' A is a lower triangular matrix.

- trans TransposeOperation
On entry, TRANS specifies the operation to be performed as follows: TRANS = 'N' or 'n' x := A*x. TRANS = 'T' or 't' x := A

^{T}*x. TRANS = 'C' or 'c' x := A^{T}*x.- diag MatrixDiagonal
On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular.

- n Int32
On entry, N specifies the order of the matrix A. N must be at least zero.

- k Int32
On entry with UPLO = 'U' or 'u', K specifies the number of super-diagonals of the matrix A. On entry with UPLO = 'L' or 'l', K specifies the number of sub-diagonals of the matrix A. K must satisfy 0 .le. K.

- a ReadOnlySpan2D<T>
A is DOUBLE PRECISION array of DIMENSION ( LDA, n ). Before entry with UPLO = 'U' or 'u', the leading ( k + 1 ) by n part of the array A must contain the upper triangular band part of the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row ( k + 1 ) of the array, the first super-diagonal starting at position 2 in row k, and so on. The top left k by k triangle of the array A is not referenced. The following program segment will transfer an upper triangular band matrix from conventional full matrix storage to band storage: DO 20, J = 1, N M = K + 1 - J DO 10, I = MAX( 1, J - K ), J A( M + I, J ) = matrix( I, J ) 10 CONTINUE 20 CONTINUE Before entry with UPLO = 'L' or 'l', the leading ( k + 1 ) by n part of the array A must contain the lower triangular band part of the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row 1 of the array, the first sub-diagonal starting at position 1 in row 2, and so on. The bottom right k by k triangle of the array A is not referenced. The following program segment will transfer a lower triangular band matrix from conventional full matrix storage to band storage: DO 20, J = 1, N M = 1 - J DO 10, I = J, MIN( N, J + K ) A( M + I, J ) = matrix( I, J ) 10 CONTINUE 20 CONTINUE Note that when DIAG = 'U' or 'u' the elements of the array A corresponding to the diagonal elements of the matrix are not referenced, but are assumed to be unity.

On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least ( k + 1 ).

- x SpanSlice<T>
X is DOUBLE PRECISION array of dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element vector x. On exit, X is overwritten with the tranformed vector x.

On entry, INCX specifies the increment for the elements of X. INCX must not be zero.

#### Type Parameters

- T

#### Usage Note

In Visual Basic and C#, you can call this method as an instance method on any object of type ILinearAlgebraOperations<T>. When you use instance method syntax to call this method, omit the first parameter. For more information, see Extension Methods (Visual Basic) or Extension Methods (C# Programming Guide).## Remarks

## Further Details:

Level 2 LinearAlgebra routine. The vector and matrix arguments are not referenced when N = 0, or M = 0 -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs.Authors: Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver, NAG Ltd.

Date: November 2011