Engineering and Scientific Subroutine Library for AIX Version 3 Release 3: Guide and Reference

SGBMV, DGBMV, CGBMV, and ZGBMV--Matrix-Vector Product for a General Band Matrix, Its Transpose, or Its Conjugate Transpose

SGBMV and DGBMV compute the matrix-vector product for either a real general band matrix or its transpose, where the general band matrix is stored in BLAS-general-band storage mode. It uses the scalars alpha and beta, vectors x and y, and general band matrix A or its transpose:

y<--betay+alphaAx

y <-- betay+alphaA^Tx

CGBMV and ZGBMV compute the matrix-vector product for either a complex general band matrix, its transpose, or its conjugate transpose, where the general band matrix is stored in BLAS-general-band storage mode. It uses the scalars alpha and beta, vectors x and y, and general band matrix A, its transpose, or its conjugate transpose:

y <-- betay+alphaAx

y <-- betay+alphaA^Tx

y <-- betay+alphaA^Hx

Table 67. Data Types

Syntax

On Entry

transa

indicates the form of matrix A to use in the computation, where:

If transa = 'N', A is used in the computation.

If transa = 'T', A^T is used in the computation.

If transa = 'C', A^H is used in the computation.

Specified as: a single character. It must be 'N', 'T', or 'C'.

m

is the number of rows in matrix A, and:

If transa = 'N', it is the length of vector y.

If transa = 'T' or 'C', it is the length of vector x.

Specified as: a fullword integer; m >= 0.

n

is the number of columns in matrix A, and:

If transa = 'N', it is the length of vector x.

If transa = 'T' or 'C', it is the length of vector y.

Specified as: a fullword integer; n >= 0.

ml

is the lower band width ml of the matrix A. Specified as: a fullword integer; ml >= 0.

mu

is the upper band width mu of the matrix A. Specified as: a fullword integer; mu >= 0.

alpha

is the scaling constant alpha. Specified as: a number of the data type indicated in Table 67.

a

is the m by n general band matrix A, stored in BLAS-general-band storage mode. It has an upper band width mu and a lower band width ml. Also:

If transa = 'N', A is used in the computation.

If transa = 'T', A^T is used in the computation.

If transa = 'C', A^H is used in the computation.

Note:

No data should be moved to form A^T or A^H; that is, the matrix A should always be stored in its untransposed form in BLAS-general-band storage mode.

Specified as: an lda by (at least) n array, containing numbers of the data type indicated in Table 67, where lda >= ml+mu+1.

lda

is the leading dimension of the array specified for a. Specified as: a fullword integer; lda > 0 and lda >= ml+mu+1.

x

is the vector x, where:

If transa = 'N', it has length n.

If transa = 'T' or 'C', it has length m.

Specified as: a one-dimensional array, containing numbers of the data type indicated in Table 67, where:

If transa = 'N', it must have at least 1+(n-1)|incx| elements.

If transa = 'T' or 'C', it must have at least 1+(m-1)|incx| elements.

incx

is the stride for vector x. Specified as: a fullword integer; incx > 0 or incx < 0.

beta

is the scaling constant beta. Specified as: a number of the data type indicated in Table 67.

y

is the vector y, where:

If transa = 'N', it has length m.

If transa = 'T' or 'C', it has length n.

Specified as: a one-dimensional array, containing numbers of the data type indicated in Table 67, where:

If transa = 'N', it must have at least 1+(m-1)|incy| elements.

If transa = 'T' or 'C', it must have at least 1+(n-1)|incy| elements.

incy

is the stride for vector y. Specified as: a fullword integer; incy > 0 or incy < 0.

On Return

y

is the vector y, containing the result of the computation, where:

If transa = 'N', it has length m.

If transa = 'T' or 'C', it has length n.

Returned as: a one-dimensional array, containing numbers of the data type indicated in Table 67.

Notes

1. For SGBMV and DGBMV, if you specify 'C' for the transa argument, it is interpreted as though you specified 'T'.
2. All subroutines accept lowercase letters for the transa argument.
3. Vector y must have no common elements with matrix A or vector x; otherwise, results are unpredictable. See Concepts.
4. To achieve optimal performance, use lda = mu+ml+1.
5. For general band matrices, if you specify ml >= m or mu >= n, ESSL assumes, only for purposes of the computation, that the lower band width is m-1 or the upper band width is n-1, respectively. However, ESSL uses the original values for ml and mu for the purposes of finding the locations of element a₁₁ and all other elements in the array specified for A, as described in General Band Matrix. For an illustration of this technique, see Example 4.
6. For a description of how a general band matrix is stored in BLAS-general-band storage mode in an array, see General Band Matrix.

Function

The possible computations that can be performed by these subroutines are described in the following sections. Varying implementation techniques are used for this computation to improve performance. As a result, accuracy of the computational result may vary for different computations.

In all the computations, general band matrix A is stored in its untransposed form in an array, using BLAS-general-band storage mode.

For SGBMV and CGBMV, intermediate results are accumulated in long precision. Occasionally, for performance reasons, these intermediate results are truncated to short precision and stored.

See references [34], [35], [38], [46], and [79]. No computation is performed if m or n is 0 or if alpha is zero and beta is one.

General Band Matrix

For SGBMV, DGBMV, CGBMV, and ZGBMV, the matrix-vector product for a general band matrix is expressed as follows:

y<--betay+alphaAx

where:

x is a vector of length n.

y is a vector of length m.

alpha is a scalar.

beta is a scalar.

A is an m by n general band matrix, having a lower band width of ml and an upper band width of mu.

Transpose of a General Band Matrix

For SGBMV, DGBMV, CGBMV, and ZGBMV, the matrix-vector product for the transpose of a general band matrix is expressed as:

y <-- betay+alphaA^Tx

where:

x is a vector of length m.

y is a vector of length n.

alpha is a scalar.

beta is a scalar.

A^T is the transpose of an m by n general band matrix A, having a lower band width of ml and an upper band width of mu.

Conjugate Transpose of a General Band Matrix

For CGBMV and ZGBMV, the matrix-vector product for the conjugate transpose of a general band matrix is expressed as follows:

y <-- betay+alphaA^Hx

where:

x is a vector of length m.

y is a vector of length n.

alpha is a scalar.

beta is a scalar.

A^H is the conjugate transpose of an m by n general band matrix A of order n, having a lower band width of ml and an upper band width of mu.

Error Conditions

Computational Errors

None

Input-Argument Errors

1. transa <> 'N', 'T', or 'C'
2. m < 0
3. n < 0
4. ml < 0
5. mu < 0
6. lda <= 0
7. lda < ml+mu+1
8. incx = 0
9. incy = 0

Example 1

This example shows how to use SGBMV to perform the computation y<--betay+alphaAx, where TRANSA is equal to 'N', and the following real general band matrix A is used in the computation. Matrix A is:

                     *                    *
                     | 1.0  1.0  1.0  0.0 |
                     | 2.0  2.0  2.0  2.0 |
                     | 3.0  3.0  3.0  3.0 |
                     | 4.0  4.0  4.0  4.0 |
                     | 0.0  5.0  5.0  5.0 |
                     *                    *

Call Statement and Input

           TRANSA M   N   ML  MU  ALPHA  A  LDA  X  INCX  BETA   Y  INCY
             |    |   |   |   |     |    |   |   |   |     |     |   |
CALL SGBMV( 'N' , 5 , 4 , 3 , 2 ,  2.0 , A , 8 , X , 1  , 10.0 , Y , 2  )

        *                    *
        |  .    .   1.0  2.0 |
        |  .   1.0  2.0  3.0 |
        | 1.0  2.0  3.0  4.0 |
A    =  | 2.0  3.0  4.0  5.0 |
        | 3.0  4.0  5.0   .  |
        | 4.0  5.0   .    .  |
        |  .    .    .    .  |
        |  .    .    .    .  |
        *                    *

X        =  (1.0, 2.0, 3.0, 4.0)
Y        =  (1.0, . , 2.0, . , 3.0, . , 4.0, . , 5.0, . )

Output

Y        =  (22.0, . , 60.0, . , 90.0, . , 120.0, . , 140.0, . )

Example 2

This example shows how to use SGBMV to perform the computation y <-- betay+alphaA^Tx, where TRANSA is equal to 'T', and the transpose of a real general band matrix A is used in the computation. It uses the same input as Example 1.

Call Statement and Input

           TRANSA M   N   ML  MU  ALPHA  A  LDA  X  INCX  BETA   Y  INCY
             |    |   |   |   |     |    |   |   |   |     |     |   |
CALL SGBMV( 'T' , 5 , 4 , 3 , 2 ,  2.0 , A , 8 , X , 1  , 10.0 , Y , 2  )

Output

Y        =  (70.0, . , 130.0, . , 140.0, . , 148.0, . )

Example 3

This example shows how to use CGBMV to perform the computation y<--betay+alphaA^Hx, where TRANSA is equal to 'C', and the complex conjugate of the following general band matrix A is used in the computation. Matrix A is:

             *                                                *
             | (1.0, 1.0)  (1.0, 1.0)  (1.0, 1.0)  (0.0, 0.0) |
             | (2.0, 2.0)  (2.0, 2.0)  (2.0, 2.0)  (2.0, 2.0) |
             | (3.0, 3.0)  (3.0, 3.0)  (3.0, 3.0)  (3.0, 3.0) |
             | (4.0, 4.0)  (4.0, 4.0)  (4.0, 4.0)  (4.0, 4.0) |
             | (0.0, 0.0)  (5.0, 5.0)  (5.0, 5.0)  (0.0, 0.0) |
             *                                                *

Call Statement and Input

           TRANSA M   N   ML  MU  ALPHA   A  LDA  X  INCX  BETA   Y  INCY
             |    |   |   |   |     |     |   |   |   |     |     |   |
CALL CGBMV( 'C' , 5 , 4 , 3 , 2 , ALPHA , A , 8 , X , 1  , BETA , Y , 2  )

        *                                                *
        |     .           .       (1.0, 1.0)  (2.0, 2.0) |
        |     .       (1.0, 1.0)  (2.0, 2.0)  (3.0, 3.0) |
        | (1.0, 1.0)  (2.0, 2.0)  (3.0, 3.0)  (4.0, 4.0) |
A    =  | (2.0, 2.0)  (3.0, 3.0)  (4.0, 4.0)  (5.0, 5.0) |
        | (3.0, 3.0)  (4.0, 4.0)  (5.0, 5.0)      .      |
        | (4.0, 4.0)  (5.0, 5.0)      .           .      |
        |     .           .           .           .      |
        |     .           .           .           .      |
        *                                                *

X        =  ((1.0, 2.0), (2.0, 3.0), (3.0, 4.0), (4.0, 5.0),
             (5.0, 6.0))
ALPHA    =  (1.0, 1.0)
BETA     =  (10.0, 0.0)
Y        =  ((1.0, 2.0), . , (2.0, 3.0), . , (3.0, 4.0), . ,
             (4.0, 5.0), . )

Output

Y        =  ((70.0, 100.0), . , (130.0, 170.0), . ,
             (140.0, 180.0), . , (148.0, 186.0), . )

Example 4

This example shows how to use SGBMV to perform the computation y<--betay+alphaAx, where ml >= m and mu  >= n, TRANSA is equal to 'N', and the following real general band matrix A is used in the computation. Matrix A is:

                       *                         *
                       | 1.0  1.0  1.0  1.0  1.0 |
                       | 2.0  2.0  2.0  2.0  2.0 |
                       | 3.0  3.0  3.0  3.0  3.0 |
                       | 4.0  4.0  4.0  4.0  4.0 |
                       *                         *

Call Statement and Input

           TRANSA M   N   ML  MU  ALPHA  A  LDA   X  INCX  BETA   Y  INCY
             |    |   |   |   |     |    |   |    |   |     |     |   |
CALL SGBMV( 'N' , 4 , 5 , 6 , 5 ,  2.0 , A , 12 , X , 1  , 10.0 , Y , 2  )

        *                         *
        |  .    .    .    .    .  |
        |  .    .    .    .   1.0 |
        |  .    .    .   1.0  2.0 |
        |  .    .   1.0  2.0  3.0 |
        |  .   1.0  2.0  3.0  4.0 |
A    =  | 1.0  2.0  3.0  4.0   .  |
        | 2.0  3.0  4.0   .    .  |
        | 3.0  4.0   .    .    .  |
        | 4.0   .    .    .    .  |
        |  .    .    .    .    .  |
        |  .    .    .    .    .  |
        |  .    .    .    .    .  |
        *                         *
 
X        =  (1.0, 2.0, 3.0, 4.0, 5.0)
Y        =  (1.0, . , 2.0, . , 3.0, . , 4.0, . )

Output

Y        =  (40.0, . , 80.0, . , 120.0, . , 160.0, . )