## blas

An interface to level 1, 2 and 3 BLAS linear algebra routines.

## Documentation

### Naming conventions for routines

Every routine in the BLAS library comes in four flavors, each prefixed by the letters S, D, C, and Z, respectively. Each letter indicates the format of input data:

• S stands for single-precision (32-bit IEEE floating point numbers),
• D stands for double-precision (64-bit IEEE floating point numbers),
• C stands for complex numbers (represented by pairs of 32-bit IEEE floating point numbers),
• Z stands for double complex numbers (represented by pairs of 64-bit IEEE floating point numbers)

In addition, each BLAS routine in this egg comes in three flavors:

• Safe, pure

Safe routines check the sizes of their input arguments. For example, if a routine is supplied arguments that indicate that an input matrix is of dimensions M-by-N, then the argument corresponding to that matrix is checked that it is of size M * N.

Pure routines do not alter their arguments in any way. A new matrix or vector is allocated for the return value of the routine.

• Safe, destructive (suffix: !)

Safe routines check the sizes of their input arguments. For example, if a routine is supplied arguments that indicate that an input matrix is of dimensions M-by-N, then the argument corresponding to that matrix is checked that it is of size M * N.

Destructive routines can modify some or all of their arguments. They are given names ending in exclamation mark. Please consult the BLAS documentation to determine which functions modify their input arguments.

• Unsafe, destructive (prefix: unsafe-, suffix: !)

Unsafe routines do not check the sizes of their input arguments. They invoke the corresponding BLAS routines directly. Unsafe routines do not have pure variants.

For example, function xGEMM (matrix-matrix multiplication) comes in the following variants:

BLAS nameSafe, pureSafe, destructiveUnsafe, destructive
SGEMMsgemmsgemm!unsafe-sgemm!
DGEMMdgemmdgemm!unsafe-dgemm!
CGEMMcgemmcgemm!unsafe-cgemm!
ZGEMMzgemmzgemm!unsafe-zgemm!

### Vector copy routines

scopy:procedure
dcopy:procedure
ccopy:procedure
zcopy:procedure

These procedures return a copy of given input SRFI-4 vector. The returned vector is allocated with the corresponding SRFI-4 constructor, and the input vector is copied to it by the corresponding BLAS copy procedure.

### BLAS level 1 routines

#### Conventions

The BLAS level 1 procedures in this egg differ from the actual routines they invoke by the position of the vector increment arguments (INCX and INCY). In this egg, these arguments are optional; the default value of INCX and INCY is 1.

In the procedure signatures below, these optional arguments are indicated by [ and ] (square brackets).

#### Apply plane rotation

srot:procedure
drot:procedure

xROT applies a plane rotation matrix to a sequence of ordered pairs: (x_i , y_i), for i = 1, 2, ..., n.

X and Y are vector of dimensions (N-1) * abs(incx) + 1 and (N-1) * abs(incy) + 1, respectively.

C and S are respectively the cosine and sine of the plane of rotation.

#### Scale vector

sscal:procedure
dscal:procedure
cscal:procedure
zscal:procedure

xSCAL scales a vector with a scalar: x := alpha * x.

#### Swap the elements of two vectors

sswap:procedure
dswap:procedure
cswap:procedure
zswap:procedure

xSWAP interchanges the elements of two vectors: x <-> y.

#### Real vector dot product

sdot:procedure
ddot:procedure

xDOT computes the dot product of two vectors of real values: dot := x'*y = \Sum_{i=1}^{n} (x_i * y_i).

#### Complex vector dot product

cdotu:procedure
zdotu:procedure

xDOTU computes the dot product of two vectors of complex values: dotu := x'*y = \Sum_{i=1}^{n} (x_i * y_i).

#### Hermitian vector dot product

cdotc:procedure
zdotc:procedure

xDOTC computes the dot product of the conjugates of two complex vectors: dotu := conjg(x')*y = \Sum_{i=1}^{n} (conjg(x_i) * y_i), for i = 1, 2, ..., n.

saxpy:procedure
daxpy:procedure
caxpy:procedure
zaxpy:procedure

xAXPY adds a scalar multiple of a vector to another vector: y := alpha * x + y.

#### Vector multiply-add with optional offset

siaxpy:procedure
diaxpy:procedure
ciaxpy:procedure
ziaxpy:procedure

xIAXPY adds a scalar multiple of a vector to another vector, where the beginning of each vector argument can be offset: y[yofs:n] := alpha * x[xofs:n] + y[yofs:n].

#### Euclidean norm of a vector

snrm2:procedure
dnrm2:procedure
cnrm2:procedure
znrm2:procedure

xNRM2 computes the Euclidean (L2) norm of a vector.

#### Sum of absolute values of the elements in a vector

sasum:procedure
dasum:procedure
casum:procedure
zasum:procedure

xASUM sums the absolute values of the elements in a vector.

#### Sum of absolute values of the elements in a vector

samax:procedure
damax:procedure
camax:procedure
zamax:procedure

xAMAX searches a vector for the first occurrence of its maximum absolute value, and returns the index of that element.

### BLAS level 2 routines

#### Conventions

The BLAS level 2 procedures in this egg differ from the actual routines they invoke by the position of the leading dimension argument (LDA) and the vector increment arguments (INCX and INCY). In this egg, these arguments are optional; the default value of LDAis the largest matrix dimension, depending on the semantics of the respective operation, and the default value of INCX and INCY is 1.

In the procedure signatures below, these optional arguments are indicated by [ and ] (square brackets).

Argument ORDER is one of RowMajor or ColMajor to indicate that the input and output matrices are in row-major or column-major form, respectively.

Where present, argument TRANS can be one of NoTrans or Trans to indicate whether the input matrix is to be transposed or not.

Where present, argument UPLO can be one of Upper or Lower to indicate whether the upper or lower triangular part of an input symmetric matrix is to referenced,or to specify the type of an input triangular matrix.

Where present, argument DIAG can be one of NonUnit or Unit to indicate whether an input triangular matrix is unit triangular or not.

sgemv:procedure
dgemv:procedure
cgemv:procedure
zgemv:procedure

xGEMV performs the matrix-vector multiply-add operation of the form y := alpha*op( A )*x + beta*y, where op( X ) is one of op( A ) = A or op( A ) = A'.

ALPHA and BETA are scalars, and A is an M x N matrix.

X is a vector of size (1 + ( N - 1 ) * abs(INCX)) when argument TRANS is NoTrans, and (1 + ( M - 1 ) * abs(INCX)) otherwise. Y is a vector of size (1 + ( M - 1 ) * abs(INCY)) when argument TRANS is NoTrans, and (1 + ( N - 1 ) * abs(INCY)) otherwise.

sgbmv:procedure
dgbmv:procedure
cgbmv:procedure
zgbmv:procedure

xGBMV performs the matrix-vector multiply-add operation of the form y := alpha*op( A )*x + beta*y, where op( X ) is one of op( A ) = A or op( A ) = A'.

ALPHA and BETA are scalars, and A is an M x N banded matrix, with KL sub-diagonals and KU super-diagonals.

X is a vector of size (1 + ( N - 1 ) * abs(INCX)) when argument TRANS is NoTrans, and (1 + ( M - 1 ) * abs(INCX)) otherwise. Y is a vector of size (1 + ( M - 1 ) * abs(INCY)) when argument TRANS is NoTrans, and (1 + ( N - 1 ) * abs(INCY)) otherwise.

chemv:procedure
zhemv:procedure

xHEMV performs the matrix-vector multiply-add operation of the form y := alpha*op( A )*x + beta*y, where op( X ) is one of op( A ) = A or op( A ) = A'.

ALPHA and BETA are scalars, and A is an N x N Hermitian matrix.

X and Y are N element vectors.

chbmv:procedure
zhbmv:procedure

xHBMV performs the matrix-vector multiply-add operation of the form y := alpha*op( A )*x + beta*y, where op( X ) is one of op( A ) = A or op( A ) = A'.

ALPHA and BETA are scalars, and A is an N x N Hermitian banded matrix, with K super-diagonals.

X and Y are N element vectors.

ssymv:procedure
dsymv:procedure

xSYMV performs matrix-vector multiply-add operation of the form y := alpha*A*x + beta*y.

ALPHA and BETA are scalars, and A is an N x N symmetric matrix.

X and Y are N element vectors.

ssbmv:procedure
dsbmv:procedure

xSBMV performs matrix-vector multiply-add operation of the form y := alpha*A*B + beta*y.

ALPHA and BETA are scalars, and A is an N x N symmetric banded matrix, with K super-diagonals.

X and Y are N element vectors.

strmv:procedure
dtrmv:procedure
ctrmv:procedure
ztrmv:procedure

xTRMV performs matrix-vector multiply-add operation of the form y := alpha*op( A )*x, where op ( A ) = A or op ( A ) = A'

ALPHA and BETA are scalars, and A is an N x N upper or lower triangular matrix.

X is a vector of length (1 + (n - 1) * abs(INCX)).

stbmv:procedure
dtbmv:procedure
ctbmv:procedure
ztbmv:procedure

xTBMV performs matrix-vector multiply-add operation of the form y := alpha*A*B + beta*y, where op ( A ) = A or op ( A ) = A'

ALPHA and BETA are scalars, and A is an N x N upper or lower triangular banded matrix, with K+1 diagonals.

X is a vector of length (1 + (n - 1) * abs(INCX)).

#### Triangular matrix equation solve

strsv:procedure
dtrsv:procedure
ctrsv:procedure
ztrsv:procedure

xTRSV solves one of the systems of equations A*x = b or A'*x = b.

ALPHA and BETA are scalars, A is a upper or lower triangular matrix, and B is a N element vector.

#### Banded triangular matrix equation solve

stbsv:procedure
dtbsv:procedure
ctbsv:procedure
ztbsv:procedure

xTBSV solves one of the systems of equations A*x = b or A'*x = b.

ALPHA and BETA are scalars, A is a upper or lower banded triangular matrix with K+1 diagonals, and B is a N element vector.

#### Rank 1 operation

sger:procedure
dger:procedure

xGER performs the rank 1 operation A := alpha*x*y' + A.

ALPHA is a scalar, X is an M element vector, Y is an N element vector, and A is an M x N matrix.

#### Rank 1 operation with optional offset

siger:procedure
diger:procedure

xIGER performs the rank 1 operation A := alpha*x[xofs:M]*y'[yofs:N] + A.

ALPHA is a scalar, X is an M element vector, Y is an N element vector, and A is an M x N matrix.

#### Rank 1 operation on complex matrices and vectors

cgeru:procedure
zgeru:procedure

xGERU performs the rank 1 operation A := alpha*x*y' + A.

ALPHA is a scalar, X is an M element vector, Y is an N element vector, and A is an M x N matrix.

#### Rank 1 operation on complex matrices and vectors

cgerc:procedure
zgerc:procedure

xGERC performs the rank 1 operation A := alpha*x*conjg(y') + A.

ALPHA is a scalar, X is an M element vector, Y is an N element vector, and A is an M x N matrix.

#### Hermitian rank 1 operation

cher:procedure
zher:procedure

xHER performs the Hermitian rank 1 operation A := alpha*x*conjg(x') + A.

ALPHA is a scalar, X is an N element vector, and A is an N x N Hermitian matrix.

#### Hermitian rank 2 operation

cher2:procedure
zher2:procedure

xHER2 performs the Hermitian rank 2 operation A := alpha*x*conjg(y') + conjg(alpha)*y*conjg(x') + A.

ALPHA is a scalar, X and Y are N element vectors, and A is an N x N Hermitian matrix.

#### Symmetric rank 1 operation

ssyr:procedure
dsyr:procedure

xSYR performs the symmetric rank 1 operation A := alpha*x*x' + A.

ALPHA is a scalar, X is an N element vector, and A is an N x N symmetric matrix.

#### Symmetric rank 2 operation

ssyr2:procedure
dsyr2:procedure

xSYR2 performs the symmetric rank 2 operation A := alpha*x*y' + alpha*y*x' + A.

ALPHA is a scalar, X and Y are N element vectors, and A is an N x N symmetric matrix.

### BLAS level 3 routines

#### Conventions

The BLAS level 3 procedures in this egg differ from the actual routines they invoke by the position of the leading dimension arguments (LDA, LDB, and LDC). In this egg, these arguments are optional, and their default values are set to the largest matrix dimension, depending on the semantics of the respective operation.

In the procedure signatures below, these optional arguments are indicated by [ and ] (square brackets).

Argument ORDER is one of RowMajor or ColMajor to indicate that the input and output matrices are in row-major or column-major form, respectively.

Where present, arguments TRANS, TRANSA, TRANSB can be one of NoTrans or Trans to indicate whether the respective input matrices are to be transposed or not.

Where present, argument SIDE can be one of Left or Right to indicate whether an input symmetric matrix appears on the left or right in the respective operation.

Where present, argument UPLO can be one of Upper or Lower to indicate whether the upper or lower triangular part of an input symmetric matrix is to referenced,or to specify the type of an input triangular matrix.

Where present, argument DIAG can be one of NonUnit or Unit to indicate whether an input triangular matrix is unit triangular or not.

sgemm:procedure
dgemm:procedure
cgemm:procedure
zgemm:procedure

xGEMM performs matrix-matrix multiply-add operation of the form C := alpha*op( A )*op( B ) + beta*C, where op( X ) is one of op( X ) = X or op( X ) = X'.

ALPHA and BETA are scalars, and A, B and C are matrices, with op( A ) an M x K matrix, op( B ) a K x N matrix and C an M x N matrix.

ssymm:procedure
dsymm:procedure
csymm:procedure
zsymm:procedure

xSYMM performs matrix-matrix multiply-add operation of the form C := alpha*A*B + beta*C or C := alpha*B*A + beta*C.

ALPHA and BETA are scalars, A is a symmetric matrix, and B and C are M x N matrices.

#### Symmetric rank k operation

ssyrk:procedure
dsyrk:procedure
csyrk:procedure
zsyrk:procedure

xSYRK performs one of the symmetric rank k operations

`{{C := alpha*A*A' + beta*C}} or {{C := alpha*A'*A + beta*C}}. `

ALPHA and BETA are scalars, A is an N x K or K x N matrix, and C is an N x N symmetric matrix.

#### Hermitian rank k operation

cherk:procedure
zherk:procedure

xHERK performs one of the hermitian rank k operations C := alpha*A*conjg(A') + beta*C or C := alpha*conjg(A')*A + beta*C.

ALPHA and BETA are scalars, A is an N x K or K x N matrix, and C is an N x N hermitian matrix.

#### Symmetric rank 2k operation

ssyr2k:procedure
dsyr2k:procedure
csyr2k:procedure
zsyr2k:procedure

xSYR2K performs one of the symmetric rank 2k operations C := alpha*A*B' + beta*C or C := alpha*B'*A + beta*C.

ALPHA and BETA are scalars, A and B are N x K or K x N matrices, and C is an N x N symmetric matrix.

#### Hermitian rank 2k operation

cher2k:procedure
zher2k:procedure

xHER2K performs one of the hermitian rank 2k operations C := alpha*A*conjg(B') + beta*C or C := alpha*conjg(B')*A + beta*C.

ALPHA and BETA are scalars, A and B are N x K or K x N matrices, and C is an N x N hermitian matrix.

#### Triangular matrix multiply

strmm:procedure
dtrmm:procedure
ctrmm:procedure
ztrmm:procedure

xTRMM performs matrix-matrix multiply operation of the form B := alpha*op( A )*B or B := alpha*B*op( A ).

ALPHA is a scalar, A is an upper or lower triangular matrix, and B is an M x N matrix.

#### Triangular matrix equation solve

strsm:procedure
dtrsm:procedure
ctrsm:procedure
ztrsm:procedure

xTRSM solves one of the matrix equations

`{{op( A )*X = alpha*B}} or {{X*op( A ) = alpha*B}}. `

op( A ) is one of op( A ) = A or op( A ) = A'.

ALPHA and BETA are scalars, A is a upper or lower triangular matrix, and B is a M x N matrix.

## Examples

```(import srfi-4 blas)

(define order ColMajor)
(define transa NoTrans)

(define m 4)
(define n 4)

(define alpha 1)
(define beta 0)

(define a				; column-major order!
(f64vector 1 2 3 4
1 1 1 1
3 4 5 6
5 6 7 8) )

(define x (f64vector 1 2 1 1))
(define y (f64vector 0 0 0 0))

(dgemv! order transa m n alpha a x beta y)

(print y)```

### Repository

This egg is hosted on the CHICKEN Subversion repository:

https://anonymous@code.call-cc.org/svn/chicken-eggs/release/5/blas

If you want to check out the source code repository of this egg and you are not familiar with Subversion, see this page.

### Version history

4.1
Fixed use of memq [thanks to Moritz Heidkamp]
4.0
3.1
Updated test script to return proper exit code
3.0
Eliminated reduntant blas: prefix from names of exported symbols
2.8
Eliminated use of noop
2.7
Switched to wiki documentation
2.6
Ported to Chicken 4
2.5
Build script updated for better cross-platform compatibility
2.4
Added iger procedures; fixed a bug in the default arguments of level 2 routines
2.3
Added iaxpy procedures; fixed a bug in the default arguments of level 1 routines
2.2
Fixed a bug in the renaming of C routines
2.1
Added eggdoc property to meta file
2.0
An overhaul of the library to introduce safe, unsafe, and pure variants of each routine
1.8
Added icopy procedures [by Ivan Raikov]
1.7
Support for Mac OS X added [by Will Farr]
1.6
Fixed bug in blas library test code
1.5
Added support for CLAPACK [by Ivan Raikov]
1.4
Added support for atlas CBLAS library [by Stu Glaser]
1.3
Tries to find a proper CBLAS library (currently only GSL)
1.2
1.1

```Copyright (c) 2003-2006, Felix L. Winkelmann

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are
met:

Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.

Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.

Neither the name of the author nor the names of its contributors may
be used to endorse or promote products derived from this software
without specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND
CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING,
BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND
FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
COPYRIGHT HOLDERS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS
OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR
TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE
USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH
DAMAGE.```