Contents

# cblas_?tbmv

Computes a matrix-vector product using a triangular band matrix.

## Syntax

Include Files
• mkl.h
Description
The
?tbmv
routines perform one of the matrix-vector operations defined as
x
:=
A
*
x
, or
x
:=
A
'*
x
, or
x
:= conjg(
A
')*
x
,
where:
x
is an
n
-element vector,
A
is an
n
-by-
n
unit, or non-unit, upper or lower triangular band matrix, with
(
k
+1)
diagonals.
Input Parameters
Layout
Specifies whether two-dimensional array storage is row-major (
CblasRowMajor
) or column-major (
CblasColMajor
).
uplo
Specifies whether the matrix
A
is an upper or lower triangular matrix:
uplo
=
CblasUpper
if
uplo
=
CblasLower
, then the matrix is low triangular.
trans
Specifies the operation:
if
trans
=
CblasNoTrans
, then
x
:=
A
*
x
;
if
trans
=
CblasTrans
, then
x
:=
A
'*
x
;
if
trans
=
CblasConjTrans
, then
x
:= conjg(
A
')*
x
.
diag
Specifies whether the matrix
A
is unit triangular:
if
diag
=
CblasUnit
then the matrix is unit triangular;
if
diag
=
CblasNonUnit
, then the matrix is not unit triangular.
n
Specifies the order of the matrix
A
. The value of
n
must be at least zero.
k
On entry with
uplo
=
CblasUpper
specifies the number of super-diagonals of the matrix
A
. On entry with
uplo
=
CblasLower
,
k
specifies the number of sub-diagonals of the matrix
a
.
The value of
k
must satisfy
0
k
.
a
Array, size
lda
*
n
.
Layout
=
CblasColMajor
:
Before entry with
uplo
=
CblasUpper
(
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
of the array, the first super-diagonal starting at position 1 in row
(
k
- 1)
, and so on.
The top left
k
by
k
triangle of the array
a
is not referenced. The following program segment transfers an upper triangular band matrix from conventional full matrix storage (
matrix
ldm
) to band storage (
a
lda
):
```

for (j = 0; j < n; j++) {
m = k - j;
for (i = max( 0, j - k); i <= j; i++) {
a[(m+i) + j*lda] = matrix[i + j*ldm];
}
}
```
Before entry with
uplo
=
CblasLower
(
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 0 of the array, the first sub-diagonal starting at position 0 in row 1
, and so on. The bottom right
k
by
k
triangle of the array
a
is not referenced. The following program segment transfers a lower triangular band matrix from conventional full matrix storage (
matrix
ldm
) to band storage (
a
lda
):
```

for (j = 0; j < n; j++) {
m = -j;
for (i = j; i < min(n, j + k + 1); i++) {
a[(m+i) + j*lda] = matrix[i + j*ldm];
}
}```
Note that when
diag
=
CblasUnit
, the elements of the array
a
corresponding to the diagonal elements of the matrix are not referenced, but are assumed to be unity.
Layout = CblasRowMajor:
Before entry with
uplo
=
CblasUpper
k
+ 1)-by-
n
part of array
a
must contain the upper triangular band part of the matrix of coefficients. The matrix must be supplied row-by-row, with the leading diagonal of the matrix in column 0 of the array, the first super-diagonal starting at position 0 in column 1, and so on. The bottom right
k
-by-
k
triangle of array
a
is not referenced.
The following program segment transfers the upper triangular part of a Hermitian band matrix from row-major full matrix storage (
matrix
ldm
) to row-major band storage (
a
lda
):
```

for (i = 0; i < n; i++) {
m = -i;
for (j = i; j < MIN(n, i+k+1); j++) {
a[(m+j) + i*lda] = matrix[j + i*ldm];
}
}
```
Before entry with
uplo
=
CblasLower
k
+ 1)-by-
n
part of array
a
must contain the lower triangular band part of the matrix of coefficients, supplied row-by-row, with the leading diagonal of the matrix in column
k
of the array, the first sub-diagonal starting at position 1 in column
k
-1, and so on. The top left
k
-by-
k
triangle of array
a
is not referenced.
The following program segment transfers the lower triangular part of a Hermitian row-major band matrix from row-major full matrix storage (
matrix
ldm
) to row-major band storage (
a
lda
):
```

for (i = 0; i < n; i++) {
m = k - i;
for (j = max(0, i-k); j <= i; j++) {
a[(m+j) + i*lda] = matrix[j + i*ldm];
}
}```
lda
a
as declared in the calling (sub)program. The value of
lda
must be at least
(
k
+ 1)
.
x
Array, size at least
(1 + (
n
- 1)*abs(
incx
))
. Before entry, the incremented array
x
must contain the
n
-element vector
x
.
incx
Specifies the increment for the elements of
x
.
The value of
incx
must not be zero.
Output Parameters
x
Overwritten with the transformed vector
x
.

#### Product and Performance Information

1

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804