Chapter 9: Basic Matrix/Vector

Operations

9.2.Other Matrix/Vector Operations

9.2.6Matrix Norm

Basic Linear Algebra Subprograms

The basic linear algebra subprograms, normally referred to as the BLAS, are routines for low-level operations such as dot products, matrix times vector, and matrix times matrix. Lawson et al. (1979) published the original set of 38 BLAS. The IMSL BLAS collection includes these 38 subprograms plus additional ones that extend their functionality. Since Dongarra et al. (1988 and 1990) published extensions to this set, it is customary to refer to the original 38 as Level 1 BLAS. The Level 1 operations are performed on one or two vectors of data. An extended set of subprograms perform operations involving a matrix and one or two vectors. These are called the Level 2 BLAS (page 1046). An additional extended set of operations on matrices is called the Level 3 BLAS (page 1046).

Users of the BLAS will often benefit from using versions of the BLAS supplied by hardware vendors, if available. This can provide for more efficient execution of many application programs. The BLAS provided by IMSL are written in FORTRAN. Those supplied by vendors may be written in other languages, such as assembler. The documentation given below for the BLAS is compatible with a vendor’s version of the BLAS that conforms to the published specifications.

Programming Notes for Level 1 BLAS

The Level 1 BLAS do not follow the usual IMSL naming conventions. Instead, the names consist of a prefix of one or more of the letters “ I,” “ S,” “ D,” “ C” and

“ Z;” a root name; and sometimes a suffix. For subprograms involving a mixture of data types, the output type is indicated by the first prefix letter. The suffix denotes a variant algorithm. The prefix denotes the type of the operation according to the following table:

Vector arguments have an increment parameter that specifies the storage space or stride between elements. The correspondence between the vectors x and y and the arguments SX and SY, and INCX and INCY is

Function subprograms SXYZ, DXYZ, page 1040, refer to a third vector argument z. The storage increment INCZ for z is defined like INCX, INCY. In the Level 1 BLAS, only positive values of INCX are allowed for operations that have a single vector argument. The loops in all of the Level 1 BLAS process the vector arguments in order of increasing i. For INCX, INCY, INCZ < 0, this implies processing in reverse storage order.

The function subprograms in the Level 1 BLAS are all illustrated by means of an assignment statement. For example, see SDOT (page 1039). Any value of a function subprogram can be used in an expression or as a parameter passed to a subprogram as long as the data types agree.

Descriptions of the Level 1 BLAS Subprograms

The set of Level 1 BLAS are summarized in Table 9.1. This table also lists the page numbers where the subprograms are described in more detail.

Specification of the Level 1 BLAS

With the definitions,

MX = max {1, 1 + (N − 1)|INCX|}

MY = max {1, 1 + (N − 1)|INCY|}

MZ = max {1, 1 + (N − 1)|INCZ|}

the subprogram descriptions assume the following FORTRAN declarations:

Since FORTRAN 77 does not include the type DOUBLE COMPLEX, subprograms with DOUBLE COMPLEX arguments are not available for all systems. Some systems use the declaration COMPLEX * 16 instead of DOUBLE COMPLEX.

In the following descriptions, the original BLAS are marked with an * in the left column.

Table 9.1: Level 1 Basic Linear Algebra Subprograms

†Higher precision accumulation used

Set a Vector to a Constant Value

CALL ISET (N, IA, IX, INCX)

CALL SSET (N, SA, SX, INCX)

CALL DSET (N, DA, DX, INCX)

CALL CSET (N, CA, CX, INCX)

CALL ZSET (N, ZA, ZX, INCX)

These subprograms set xL ¬ a for i = 1, 2, ¼, N. If N £ 0, then the subprograms return immediately.

Copy a Vector

CALL ICOPY (N, IX, INCX, IY, INCY)

*CALL SCOPY (N, SX, INCX, SY, INCY)

*CALL DCOPY (N, DX, INCX, DY, INCY)

*CALL CCOPY (N, CX, INCX, CY, INCY) CALL ZCOPY (N, ZX, INCX, ZY, INCY)

These subprograms set yL ¬ xL for i = 1, 2, ¼, N. If N £ 0, then the subprograms return immediately.

Scale a Vector

*CALL SSCAL (N, SA, SX, INCX)

*CALL DSCAL (N, DA, DX, INCX)

*CALL CSCAL (N, CA, CX, INCX) CALL ZSCAL (N, ZA, ZX, INCX)

*CALL CSSCAL (N, SA, CX, INCX) CALL ZDSCAL (N, DA, ZX, INCX)

These subprograms set xL ¬ axL for i = 1, 2, ¼, N. If N £ 0, then the subprograms return immediately. CAUTION: For CSSCAL and ZDSCAL, the scalar quantity a is real and the vector x is complex.

Multiply a Vector by a Constant

CALL SVCAL (N, SA, SX, INCX, SY, INCY)

CALL DVCAL (N, DA, DX, INCX, DY, INCY)

CALL CVCAL (N, CA, CX, INCX, CY, INCY)

CALL ZVCAL (N, ZA, ZX, INCX, ZY, INCY)

CALL CSVCAL (N, SA, CX, INCX, CY, INCY)

CALL ZDVCAL (N, DA, ZX, INCX, ZY, INCY)

These subprograms set yL ¬ axL for i = 1, 2, ¼, N. If N £ 0, then the subprograms return immediately. CAUTION: For CSVCAL and ZDVCAL, the scalar quantity a is real and the vector x is complex.

Add a Constant to a Vector

CALL IADD (N, IA, IX, INCX)

CALL SADD (N, SA, SX, INCX)

CALL DADD (N, DA, DX, INCX)

CALL CADD (N, CA, CX, INCX)

CALL ZADD (N, ZA, ZX, INCX)

These subprograms set xL ¬ xL + a for i = 1, 2, ¼, N. If N £ 0, then the subprograms return immediately.

Subtract a Vector from a Constant

CALL ISUB (N, IA, IX, INCX)

CALL SSUB (N, SA, SX, INCX)

CALL DSUB (N, DA, DX, INCX)

CALL CSUB (N, CA, CX, INCX)

CALL ZSUB (N, ZA, ZX, INCX)

These subprograms set xL ¬ a - xL for i = 1, 2, ¼, N. If N £ 0, then the subprograms return immediately.

Constant Times a Vector Plus a Vector

*CALL SAXPY (N, SA, SX, INCX, SY, INCY)

*CALL DAXPY (N, DA, DX, INCX, DY, INCY)

*CALL CAXPY (N, CA, CX, INCX, CY, INCY) CALL ZAXPY (N, ZA, ZX, INCX, ZY, INCY)

These subprograms set yL ¬ axL + yL for i = 1, 2, ¼, N. If N £ 0, then the subprograms return immediately.

Swap Two Vectors

CALL ISWAP (N, IX, INCX, IY, INCY)

*CALL SSWAP (N, SX, INCX, SY, INCY)

*CALL DSWAP (N, DX, INCX, DY, INCY)

*CALL CSWAP (N, CX, INCX, CY, INCY) CALL ZSWAP (N, ZX, INCX, ZY, INCY)

These subprograms perform the exchange yL « xL for i = 1, 2, ¼, N. If N £ 0, then the subprograms return immediately.

Dot Product

*SW = SDOT (N, SX, INCX, SY, INCY)

*DW = DDOT (N, DX, INCX, DY, INCY)

*CW = CDOTU (N, CX, INCX, CY, INCY)

*CW = CDOTC (N, CX, INCX, CY, INCY) ZW = ZDOTU (N, ZX, INCX, ZY, INCY) ZW = ZDOTC (N, ZX, INCX, ZY, INCY)

The function subprograms SDOT, DDOT, CDOTU, and ZDOTU compute

åiN=1 xi yi

The function subprograms CDOTC and ZDOTC compute

åiN=1 xi yi

The suffix C indicates that the complex conjugates of xL are used. The suffix U indicates that the unconjugated values of xL are used. If N ≤ 0, then the subprograms return zero.

Dot Product with Higher Precision Accumulation

The function subprogram DSDOT computes

åiN=1 xi yi

using double precision accumulation. The function subprograms CZDOTU and ZQDOTU compute

åiN=1 xi yi

using double and quadruple complex accumulation, respectively. The function subprograms CZDOTC and ZQDOTC compute

åiN=1 xi yi

using double and quadruple complex accumulation, respectively. If N ≤ 0, then the subprograms return zero.

Constant Plus Dot Product with Higher Precision Accumulation

*SW = SDSDOT (N, SA, SX, INCX, SY, INCY) DW = DQDDOT (N, DA, DX, INCX, DY, INCY) CW = CZCDOT (N, CA, CX, INCX, CY, INCY) CW = CZUDOT (N, CA, CX, INCX, CY, INCY)

ZW = ZQCDOT (N, ZA, ZX, INCX, ZY, INCY)

ZW = ZQUDOT (N, ZA, ZX, INCX, ZY, INCY)

The function subprograms SDSDOT, DQDDOT, CZUDOT, and ZQUDOT compute

a + åiN=1 xi yi

using higher precision accumulation where SDSDOT uses double precision accumulation, DQDDOT uses quadruple precision accumulation, CZUDOT uses double complex accumulation, and ZQUDOT uses quadruple complex accumulation. The function subprograms CZCDOT and ZQCDOT compute

a + åiN=1 xi yi

using double complex and quadruple complex accumulation, respectively. If N £ 0, then the subprograms return zero.

Dot Product Using the Accumulator

The variable DACC, a double precision array of length two, is used as a quadruple precision accumulator. DZACC, a double precision array of length four, is its complex analog. The function subprograms, with a name ending in I, initialize DACC to zero. All of the function subprograms then compute

DACC + b + åiN=1 xi yi

and store the result in DACC. The result, converted to the precision of the function, is also returned as the function value. If N £ 0, then the function subprograms return zero.

Hadamard Product

CALL SHPROD (N, SX, INCX, SY, INCY, SZ, INCZ)

CALL DHPROD (N, DX, INCX, DY, INCY, DZ, INCZ)

These subprograms set zL ¬ xLyL for i = 1, 2, ¼, N. If N £ 0, then the subprograms return immediately.

Triple Inner Product

SW = SXYZ (N, SX, INCX, SY, INCY, SZ, INCZ)

DW = DXYZ (N, DX, INCX, DY, INCY, DZ, INCZ)

These function subprograms compute

åiN=1 xi yi zi

If N ≤ 0 then the subprograms return zero.

Sum of the Elements of a Vector

IW = ISUM (N, IX, INCX)

SW = SSUM (N, SX, INCX)

DW = DSUM (N, DX, INCX)

These function subprograms compute

åiN=1 xi

If N ≤ 0, then the subprograms return zero.

Sum of the Absolute Values of the Elements of a Vector

*SW = SASUM (N, SX, INCX)

*DW = DASUM (N, DX, INCX)

*SW = SCASUM (N, CX, INCX) DW = DZASUM (N, ZX, INCX)

The function subprograms SASUM and DASUM compute

åiN=1 xi

The function subprograms SCASUM and DZASUM compute

åiN=1 xi + xi

If N ≤ 0, then the subprograms return zero. CAUTION: For SCASUM and DZASUM, the function subprogram returns a real value.

Euclidean or l 2 Norm of a Vector

*SW = SNRM2 (N, SX, INCX)

*DW = DNRM2 (N, DX, INCX)

*SW = SCNRM2 (N, CX, INCX) DW = DZNRM2 (N, ZX, INCX)

These function subprograms compute

åiN=1 xi 2 1 2

If N ≤ 0, then the subprograms return zero. CAUTION: For SCNRM2 and DZNRM2, the function subprogram returns a real value.

Product of the Elements of a Vector

SW = SPRDCT (N, SX, INCX)

DW = DPRDCT (N, DX, INCX)

These function subprograms compute

∏iN=1 xi

If N ≤ 0, then the subprograms return zero.

Index of Element Having Minimum Value

IW = IIMIN (N, IX, INCX)

IW = ISMIN (N, SX, INCX)

IW = IDMIN (N, DX, INCX)

These function subprograms compute the smallest index i such that xL = min1‹M‹1 xM. If N ≤ 0, then the subprograms return zero.

Index of Element Having Maximum Value

IW = IIMAX (N, IX, INCX)

IW = ISMAX (N, SX, INCX)

IW = IDMAX (N, DX, INCX)

These function subprograms compute the smallest index i such that xL = max1‹M‹1 xM. If N ≤ 0, then the subprograms return zero.

Index of Element Having Minimum Absolute Value

IW = ISAMIN (N, SX, INCX)

IW = IDAMIN (N, DX, INCX)

IW = ICAMIN (N, CX, INCX)

IW = IZAMIN (N, ZX, INCX)

The function subprograms ISAMIN and IDAMIN compute the smallest index i such that |xL| = min1‹M‹1 |xM|. The function subprograms ICAMIN and IZAMIN compute the smallest index i such that

xi + xi = min x j + x j

1≤ j≤ N

If N ≤ 0, then the subprograms return zero.

Index of Element Having Maximum Absolute Value

*IW = ISAMAX (N, SX, INCX)

*IW = IDAMAX (N, DX, INCX)

*IW = ICAMAX (N, CX, INCX) IW = IZAMAX (N, ZX, INCX)

The function subprograms ISAMAX and IDAMAX compute the smallest index i such that |xL| = max1‹M‹1 |xM|. The function subprograms ICAMAX and IZAMAX compute the smallest index i such that

xi + xi = max x j + x j

1≤ j≤ N

If N ≤ 0, then the subprograms return zero.

Construct a Givens Plane Rotation

*CALL SROTG (SA, SB, SC, SS)

*CALL DROTG (SA, SB, SC, SS)

Given the values a and b, these subprograms compute

The introduction of σ is not essential to the computation of the Givens rotation matrix; but its use permits later stable reconstruction of c and s from just one stored number, an idea due to Stewart (1976). For this purpose, the subprogram also computes

%s if s < c or c = 0 z = &

'1 / c if 0 < c £ s

In addition to returning c and s, the subprograms return r overwriting a, and z overwriting b.

Reconstruction of c and s from z can be done as follows:

If z = 1, then set c = 0 and s = 1

If |z| < 1, then set

c = 1 - z2 and s = z

If |z| > 1, then set

c = 1 / z and s = 1- c2

Apply a Plane Rotation

*CALL SROT (N, SX, INCX, SY, INCY, SC, SS)

*CALL DROT (N, DX, INCX, DY, INCY, DC, DS)

If N ≤ 0, then the subprograms return immediately. CAUTION: For CSROT and ZDROT, the scalar quantities c and s are real, and x and y are complex.

Construct a Modified Givens Transformation

*CALL SROTMG (SD1, SD2, SX1, SY1, SPARAM)

*CALL DROTMG (DD1, DD2, DX1, DY1, DPARAM)

The input quantities d1, d2, x1 and y1 define a 2-vector [w1, z1]7 by the following:

The subprograms determine the modified Givens rotation matrix H that transforms y1, and thus, z1 to zero. They also replace d1, d2 and x1 with

A representation of this matrix is stored in the array SPARAM or DPARAM. The form of the matrix H is flagged by PARAM(1).

PARAM(1) = 1. In this case,

The elements PARAM(3) and PARAM(4) are not changed.

PARAM(1) = 0. In this case,

d1x12 > d2 y12

and

The elements PARAM(2) and PARAM(5) are not changed.

PARAM(1) = −1. In this case, rescaling was done and

PARAM(1) = −2. In this case, H = I where I is the identity matrix. The elements PARAM(2), PARAM(3), PARAM(4) and PARAM(5) are not changed.

The values of d1, d2 and x1 are changed to represent the effect of the transformation. The quantity y1, which would be zeroed by the transformation, is left unchanged.

The input value of d1 should be nonnegative, but d2 can be negative for the purpose of removing data from a least-squares problem.

See Lawson et al. (1979) for further details.

Apply a Modified Givens Transformation

*CALL SROTM (N, SX, INCX, SY, INCY, SPARAM)

*CALL DROTM (N, DX, INCX, DY, INCY, DPARAM) CALL CSROTM (N, CX, INCX, CY, INCY, SPARAM) CALL ZDROTM (N, ZX, INCX, ZY, INCY, DPARAM)

If N ≤ 0 or if PARAM(1) = −2.0, then the subprograms return immediately. CAUTION: For CSROTM and ZDROTM, the scalar quantities PARAM(*) are real and x and y are complex.

Programming Notes for Level 2 and Level 3 BLAS

For definitions of the matrix data structures used in the discussion below, see page 1206. The Level 2 and Level 3 BLAS, like the Level 1 BLAS, do not follow the IMSL naming conventions. Instead, the names consist of a prefix of one of the letters “ S,” “ D,” “ C” or “ Z.” Next is a root name denoting the kind of matrix. This

is followed by a suffix indicating the type of the operation.1 The prefix denotes the type of operation according to the following table:

The root names for the kind of matrix:

1IMSL does not support the Packed Symmetric, Packed-Hermitian, or PackedTriangular data structures, with respective root names SP, HP or TP, nor any extended precision versions of the Level 2 BLAS.

The specifications of the operations are provided by subprogram arguments of CHARACTER*1 data type. Both lower and upper case of the letter have the same meaning:

Note: See the “Triangular Mode” section in the Reference Material (page 1208) for definitions of these terms.

Descriptions of the Level 2 and Level 3 BLAS

The subprograms for Level 2 and Level 3 BLAS that perform operations involving the expression by or bC do not require that the contents of y or C be defined when b = 0. In that case, the expression by or bC is defined to be zero. Note that for the _GEMV and _GBMV subprograms, the dimensions of the vectors x and y are implied by the specification of the operation. If TRANS = ’N’ , the dimension of y is m; if TRANS = ’T’ or = ’C’, the dimension of y is n. The Level 2 and Level 3 BLAS are summarized in Table 9.2. This table also lists the page numbers where the subprograms are described in more detail.

Specification of the Level 2 BLAS

Type and dimension for variables occurring in the subprogram specifications are

There is a lower bound on the leading dimension LDA. It must be ³ the number of rows in the matrix that is contained in this array. Vector arguments have an increment parameter that specifies the storage space or stride between elements. The correspondence between the vector x, y and the arguments SX, SY and INCX,

INCY is

In the Level 2 BLAS, only nonzero values of INCX, INCY are allowed for operations that have vector arguments. The Level 3 BLAS do not refer to INCX,

INCY.

Specification of the Level 3 BLAS

Type and dimension for variables occurring in the subprogram specifications are

Each of the integers K, M, N must be ³ 0. It is an error if any of them are < 0. If any of them are = 0, the subprograms return immediately. There are lower bounds on the leading dimensions LDA, LDB, LDC. Each must be ³ the number of rows in the matrix that is contained in this array.

Table 9.2: Level 2 and 3 Basic Linear Algebra Subprograms

Matrix–Vector Multiply, General

CALL SGEMV (TRANS, M, N, SALPHA, SA, LDA, SX, INCX, SBETA,SY, INCY)

CALL DGEMV (TRANS, M, N, DALPHA, DA, LDA, DX, INCX, DBETA, DY, INCY)

CALL CGEMV (TRANS, M, N, CALPHA, CA, LDA, CX, INCX, CBETA, CY, INCY)

CALL ZGEMV (TRANS, M, N, ZALPHA, ZA, LDA, ZX, INCX, ZBETA, ZY, INCY)

For all data types, A is an M × N matrix. These subprograms set y to one of the expressions: y ← αAx + βy, y ← αA7x + βy, or for complex data,

y ← αA T + βy

The character flag TRANS determines the operation.

Matrix–Vector Multiply, Banded

CALL SGBMV (TRANS, M, N, NLCA, NUCA SALPHA, SA, LDA, SX, INCX, SBETA,SY, INCY)

CALL DGBMV (TRANS, M, N, NLCA, NUCA DALPHA, DA, LDA, DX, INCX, DBETA,DY, INCY)

CALL CGBMV (TRANS, M, N, NLCA, NUCA CALPHA, CA, LDA, CX, INCX, CBETA,CY, INCY)

CALL ZGBMV (TRANS, M, N, NLCA, NUCA ZALPHA, ZA, LDA, ZX, INCX, ZBETA,ZY, INCY)

For all data types, A is an M × N matrix with NLCA lower codiagonals and NUCA upper codiagonals. The matrix is stored in band storage mode. These

subprograms set y to one of the expressions: y ← αAx + βy, y ← αA7x + βy, or for complex data,

y ← αA T x + βy

The character flag TRANS determines the operation.

Matrix-Vector Multiply, Hermitian

CALL CHEMV (UPLO, N, CALPHA, CA, LDA, CX, INCX, CBETA, CY,INCY)

CALL ZHEMV (UPLO, N, ZALPHA, ZA, LDA, ZX, INCX, ZBETA, ZY, INCY)

For all data types, A is an N × N matrix. These subprograms set y ← αAx + βy where A is an Hermitian matrix. The matrix A is either referenced using its upper or lower triangular part. The character flag UPLO determines the part used.

Matrix-Vector Multiply, Hermitian and Banded

CALL CHBMV (UPLO, N, NCODA, CALPHA, CA, LDA, CX, INCX, CBETA, CY,INCY)

CALL ZHBMV (UPLO, N, NCODA, ZALPHA, ZA, LDA, ZX, INCX, ZBETA, ZY,INCY)

For all data types, A is an N × N matrix with NCODA codiagonals. The matrix is stored in band Hermitian storage mode. These subprograms set y ← αAx + βy. The matrix A is either referenced using its upper or lower triangular part. The character flag UPLO determines the part used.

Matrix-Vector Multiply, Symmetric and Real

CALL SSYMV (UPLO, N, SALPHA, SA, LDA, SX, INCX, SBETA, SY, INCY)

CALL DSYMV (UPLO, N, DALPHA, DA, LDA, DX, INCX, DBETA, DY, INCY)

For all data types, A is an N × N matrix. These subprograms set y ← αAx + βy where A is a symmetric matrix. The matrix A is either referenced using its upper or lower triangular part. The character flag UPLO determines the part used.

Matrix-Vector Multiply, Symmetric and Banded

CALL SSBMV (UPLO, N, NCODA, SALPHA, SA, LDA, SX, INCX, SBETA, SY,INCY)

CALL DSBMV (UPLO, N, NCODA, DALPHA, DA, LDA, DX, INCX, DBETA, DY,INCY)

For all data types, A is an N × N matrix with NCODA codiagonals. The matrix is stored in band symmetric storage mode. These subprograms set y ← αAx + βy. The matrix A is either referenced using its upper or lower triangular part. The character flag UPLO determines the part used.

Matrix-Vector Multiply, Triangular

CALL STRMV (UPLO, TRANS, DIAG, N, SA, LDA, SX, INCX) CALL DTRMV (UPLO, TRANS, DIAG, N, DA, LDA, DX, INCX)

CALL CTRMV (UPLO, TRANS, DIAG, N, CA, LDA, CX, INCX) CALL ZTRMV (UPLO, TRANS, DIAG, N, ZA, LDA, ZX, INCX)

For all data types, A is an N × N triangular matrix. These subprograms set x to one of the expressions: x ← Ax, x ←A7x, or for complex data,

x ← A T x

The matrix A is either referenced using its upper or lower triangular part and is unit or nonunit triangular. The character flags UPLO, TRANS, and DIAG determine the part of the matrix used and the operation performed.

Matrix-Vector Multiply, Triangular and Banded

For all data types, A is an N × N matrix with NCODA codiagonals. The matrix is stored in band triangular storage mode. These subprograms set x to one of the

expressions: x ← Ax, x ← A7x, or for complex data,

x ← A T x

The matrix A is either referenced using its upper or lower triangular part and is unit or nonunit triangular. The character flags UPLO, TRANS, and DIAG determine the part of the matrix used and the operation performed.

Matrix-Vector Solve, Triangular

CALL STRSV (UPLO, TRANS, DIAG, N, SA, LDA, SX, INCX) CALL DTRSV (UPLO, TRANS, DIAG, N, DA, LDA, DX, INCX) CALL CTRSV (UPLO, TRANS, DIAG, N, CA, LDA, CX, INCX) CALL ZTRSV (UPLO, TRANS, DIAG, N, ZA, LDA, ZX, INCX)

For all data types, A is an N × N triangular matrix. These subprograms solve x for

one of the expressions: x ← A-1 x, x ← (A-1 )7x, or for complex data,

x ← 3 A T 8−1 x

The matrix A is either referenced using its upper or lower triangular part and is unit or nonunit triangular. The character flags UPLO, TRANS, and DIAG determine the part of the matrix used and the operation performed.

Matrix-Vector Solve, Triangular and Banded

CALL STBSV (UPLO, TRANS, DIAG, N, NCODA, SA, LDA, SX, INCX) CALL DTBSV (UPLO, TRANS, DIAG, N, NCODA, DA, LDA, DX, INCX) CALL CTBSV (UPLO, TRANS, DIAG, N, NCODA, CA, LDA, CX, INCX) CALL ZTBSV (UPLO, TRANS, DIAG, N, NCODA, ZA, LDA, ZX, INCX)

For all data types, A is an N × N triangular matrix with NCODA codiagonals. The matrix is stored in band triangular storage mode. These subprograms solve x for one of the expressions: x ← A-1 x, x ← (AT )-1x, or for complex data,

x ← 3 A T 8−1 x

The matrix A is either referenced using its upper or lower triangular part and is unit or nonunit triangular. The character flags UPLO, TRANS, and DIAG determine the part of the matrix used and the operation performed.

Rank-One Matrix Update, General and Real

CALL SGER (M, N, SALPHA, SX, INCX, SY, INCY, SA, LDA) CALL DGER (M, N, DALPHA, DX, INCX, DY, INCY, DA, LDA

For all data types, A is an M × N matrix. These subprograms set A ← A + αxy7.

Rank-One Matrix Update, General, Complex, and Transpose

CALL CGERU (M, N, CALPHA, CX, INCX, CY, INCY, CA, LDA) CALL ZGERU (M, N, ZALPHA, ZX, INCX, ZY, INCY, ZA, LDA)

For all data types, A is an M × N matrix. These subprograms set A ← A + αxy7.

Rank-One Matrix Update, General, Complex, and Conjugate

Transpose

CALL CGERC (M, N, CALPHA, CX, INCX, CY, INCY, CA, LDA) CALL ZGERC (M, N, ZALPHA, ZX, INCX, ZY, INCY, ZA, LDA)

For all data types, A is an M × N matrix. These subprograms set

A ← A + αxy T

Rank-One Matrix Update, Hermitian and Conjugate Transpose

CALL CHER (UPLO, N, SALPHA, CX, INCX, CA, LDA) CALL ZHER (UPLO, N, DALPHA, ZX, INCX, ZA, LDA)

For all data types, A is an N × N matrix. These subprograms set

A ← A + αxx T

where A is Hermitian. The matrix A is either referenced by its upper or lower triangular part. The character flag UPLO determines the part used. CAUTION: Notice the scalar parameter α is real, and the data in the matrix and vector are complex.

Rank-Two Matrix Update, Hermitian and Conjugate Transpose

CALL CHER2 (UPLO, N, CALPHA, CX, INCX, CY, INCY, CA, LDA) CALL ZHER2 (UPLO, N, ZALPHA, ZX, INCX, ZY, INCY, ZA, LDA)

For all data types, A is an N × N matrix. These subprograms set

A ← A + αxy T + αyx T

where A is an Hermitian matrix. The matrix A is either referenced by its upper or lower triangular part. The character flag UPLO determines the part used.

Rank-One Matrix Update, Symmetric and Real

CALL SSYR (UPLO, N, SALPHA, SX, INCX, SA, LDA)

CALL DSYR (UPLO, N, DALPHA, DX, INCX, DA, LDA)

For all data types, A is an N × N matrix. These subprograms set A ← A + αxx7 where A is a symmetric matrix. The matrix A is either referenced by its upper or lower triangular part. The character flag UPLO determines the part used.

Rank-Two Matrix Update, Symmetric and Real

CALL SSYR2 (UPLO, N, SALPHA, SX, INCX, SY, INCY, SA, LDA) CALL DSYR2 (UPLO, N, DALPHA, DX, INCX, DY, INCY, DA, LDA)

For all data types, A is an N × N matrix. These subprograms set A ← A + αxy7 +

αyx7 where A is a symmetric matrix. The matrix A is referenced by its upper or lower triangular part. The character flag UPLO determines the part used.

Matrix-Matrix Multiply, General

CALL SGEMM (TRANSA, TRANSB, M, N, K, SALPHA, SA, LDA, SB, LDB, SBETA, SC, LDC)

CALL DGEMM (TRANSA, TRANSB, M, N, K, DALPHA, DA, LDA, DB, LDB, DBETA, DC, LDC)

CALL CGEMM (TRANSA, TRANSB, M, N, K, CALPHA, CA, LDA, CB, LDB, CBETA, CC, LDC)

CALL ZGEMM (TRANSA, TRANSB, M, N, K, ZALPHA, ZA, LDA, ZB, LDB, ZBETA, ZC, LDC)

For all data types, these subprograms set C0œ 1 to one of the expressions:

C ← αAB + βC, C ← αAT B + βC, C ← αABT + βC, C ← αAT BT + βC,

or for complex data, C ← αAB T + βC, C ← αA T B + βC, C ← αAT B T + βC,

C ← αA T BT + βC, C ← αA T B T + βC

The character flags TRANSA and TRANSB determine the operation to be performed. Each matrix product has dimensions that follow from the fact that C has dimension M × N.

Matrix-Matrix Multiply, Symmetric

CALL SSYMM (SIDE, UPLO, M, N, SALPHA, SA, LDA, SB, LDB, SBETA, SC, LDC)

CALL DSYMM (SIDE, UPLO, M, N, DALPHA, DA, LDA, DB, LDB, DBETA, DC, LDC)

CALL CSYMM (SIDE, UPLO, M, N, CALPHA, CA, LDA, CB, LDB, CBETA, CC, LDC)

CALL ZSYMM (SIDE, UPLO, M, N, ZALPHA, ZA, LDA, ZB, LDB, ZBETA, ZC, LDC)

For all data types, these subprograms set C0 œ 1 to one of the expressions: C ← α

AB + βC or C ← αBA + βC, where A is a symmetric matrix. The matrix A is referenced either by its upper or lower triangular part. The character flags SIDE and UPLO determine the part of the matrix used and the operation performed.

Matrix-Matrix Multiply, Hermitian

CALL CHEMM (SIDE, UPLO, M, N, CALPHA, CA, LDA, CB, LDB, CBETA, CC, LDC)

CALL ZHEMM (SIDE, UPLO, M, N, ZALPHA, ZA, LDA, ZB, LDB, ZBETA, ZC, LDC)

For all data types, these subprograms set C0 œ 1 to one of the expressions: C ← α

AB + βC or C ← αBA + βC, where A is an Hermitian matrix. The matrix A is referenced either by its upper or lower triangular part. The character flags SIDE and UPLO determine the part of the matrix used and the operation performed.

Rank-k Update, Symmetric

CALL SSYRK (UPLO, TRANS, N, K, SALPHA, SA, LDA, SBETA, SC, LDC)

CALL DSYRK (UPLO, TRANS, N, K, DALPHA, DA, LDA, DBETA, DC, LDC)

CALL CSYRK (UPLO, TRANS, N, K, CALPHA, CA, LDA, CBETA, CC, LDC)

CALL ZSYRK (UPLO, TRANS, N, K, ZALPHA, ZA, LDA, ZBETA, ZC, LDC)

For all data types, these subprograms set C0 œ 1 to one of the expressions: C ← α

AA7 + βC or C ← αA7A + βC. The matrix C is referenced either by its upper or lower triangular part. The character flags UPLO and TRANS determine the part of the matrix used and the operation performed. In subprogram CSYRK and ZSYRK, only values ’N’ or ’T’ are allowed for TRANS; ’C’is not acceptable.

Rank-k Update, Hermitian

CALL CHERK (UPLO, TRANS, N, K, SALPHA, CA, LDA, SBETA, CC, LDC)

CALL ZHERK (UPLO, TRANS, N, K, DALPHA, ZA, LDA, DBETA, ZC, LDC)

For all data types, these subprograms set C1 œ 1 to one of the expressions:

C ← αAA T + βC or C ← αA T A + βC

The matrix C is referenced either by its upper or lower triangular part. The character flags UPLO and TRANS determine the part of the matrix used and the operation performed. CAUTION: Notice the scalar parameters α and β are real, and the data in the matrices are complex. Only values ’N’or ’C’are allowed for TRANS; ’T’is not acceptable.

Rank-2k Update, Symmetric

CALL SSYR2K (UPLO, TRANS, N, K, SALPHA, SA, LDA, SB, LDB, SBETA, SC, LDC)

CALL DSYR2K (UPLO, TRANS, N, K, DALPHA, DA, LDA, DB, LDB, DBETA, DC, LDC)

CALL CSYR2K (UPLO, TRANS, N, K, CALPHA, CA, LDA, CB, LDB, CBETA, CC, LDC)

CALL ZSYR2K (UPLO, TRANS, N, K, ZALPHA, ZA, LDA, ZB, LDB, ZBETA, ZC, LDC)

For all data types, these subprograms set C1œ1 to one of the expressions:

C ← αABT + αβAT + βC or C ← αAT B + αBT A + βC

The matrix C is referenced either by its upper or lower triangular part. The character flags UPLO and TRANS determine the part of the matrix used and the operation performed. In subprogram CSYR2K and ZSYR2K, only values ’N’or ’T’ are allowed for TRANS; ’C’is not acceptable.

Rank-2k Update, Hermitian

CALL CHER2K (UPLO, TRANS, N, K, CALPHA, CA, LDA, CB, LDB, SBETA, CC, LDC)

CALL ZHER2K (UPLO, TRANS, N, K, ZALPHA, ZA, LDA, ZB, LDB, DBETA, ZC, LDC)

For all data types, these subprograms set C1œ1 to one of the expressions:

C ← αAB T + αBA T + βC or C ← αA T B + αB T A + βC

The matrix C is referenced either by its upper or lower triangular part. The character flags UPLO and TRANS determine the part of the matrix used and the operation performed. CAUTION: Notice the scalar parameter β is real, and the data in the matrices are complex. In subprogram CHER2K and ZHER2K, only values ’N’ or ’C’are allowed for TRANS; ’T’is not acceptable.

Matrix-Matrix Multiply, Triangular

CALL STRMM (SIDE, UPLO, TRANSA, DIAG, M, N, SALPHA, SA, LDA, SB, LDB)

CALL DTRMM (SIDE, UPLO, TRANSA, DIAG, M, N, DALPHA, DA, LDA, DB, LDB)

CALL CTRMM (SIDE, UPLO, TRANSA, DIAG, M, N, CALPHA, CA, LDA, CB,LDB)

CALL ZTRMM (SIDE, UPLO, TRANSA, DIAG, M, N, ZALPHA, ZA, LDA, ZB, LDB)

For all data types, these subprograms set B0œ 1 to one of the_expressions:

B ← αAB, B ← αAT B, B ← αBA, B ← αBAT , or for complex data, B ← αA T B, or B ← αBA T

where A is a triangular matrix. The matrix A is either referenced using its upper or lower triangular part and is unit or nonunit triangular. The character flags SIDE, UPLO, TRANSA, and DIAG determine the part of the matrix used and the operation performed.

Matrix-Matrix Solve, Triangular

CALL STRSM (SIDE, UPLO, TRANSA, DIAG, M, N, SALPHA, SA,

LDA, SB, LDB)

CALL DTRSM (SIDE, UPLO, TRANSA, DIAG, M, N, DALPHA, DA,

LDA, DB, LDB)

CALL CTRSM (SIDE, UPLO, TRANSA, DIAG, M, N, CALPHA, CA,

LDA, CB, LDB)

CALL ZTRSM (SIDE, UPLO, TRANSA, DIAG, M, N, ZALPHA, ZA,

LDA, ZB, LDB)

For all data types, these subprograms set B0œ 1 to one of the expressions:

B ← αA−1B, B ← αBA−1, B ← α3 A−18T B, B ← αB3 A−18T , or for complex data, B ← α3 A T 8−1 B, or B ← αB3A T 8−1

where A is a triangular matrix. The matrix A is either referenced using its upper or lower triangular part and is unit or nonunit triangular. The character flags SIDE, UPLO, TRANSA, and DIAG determine the part of the matrix used and the operation performed.

Other Matrix/Vector Operations

This section describes a set of routines for matrix/vector operations. The matrix copy and conversion routines are summarized by the following table:

The matrix multiplication routines are summarized as follows:

The matrix norm routines are summarized as follows:

CRGRG/DCRGRG (Single/Double precision)

Copy a real general matrix.

Usage

CALL CRGRG (N, A, LDA, B, LDB)

Arguments

N — Order of the matrices. (Input)

A — Matrix of order N. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement of the calling program. (Input)

B — Matrix of order N containing a copy of A. (Output)

LDB — Leading dimension of B exactly as specified in the dimension statement of the calling program. (Input)

Algorithm

The routine CRGRG copies the real N × N general matrix A into the real N × N general matrix B.

CCGCG/DCCGCG (Single/Double precision)

Copy a complex general matrix.

Usage

CALL CCGCG (N, A, LDA, B, LDB)

Arguments

N — Order of the matrices A and B. (Input)

A — Complex matrix of order N. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement of the calling program. (Input)

B — Complex matrix of order N containing a copy of A. (Output)

LDB — Leading dimension of B exactly as specified in the dimension statement of the calling program. (Input)

Algorithm

The routine CCGCG copies the complex N × N general matrix A into the complex N × N general matrix B.

Example

A complex 3 × 3 general matrix is copied into another complex 3 × 3 general matrix.

DATA A/(0.0,0.0), 2*(-1.0,-1.0), (1.0,1.0), (0.0,0.0),

&(-1.0,-1.0), 2*(1.0,1.0), (0.0,0.0)/

Output

CRBRB/DCRBRB (Single/Double precision)

Copy a real band matrix stored in band storage mode.

Usage

CALL CRBRB (N, A, LDA, NLCA, NUCA, B, LDB, NLCB, NUCB)

LDA — Leading dimension of A exactly as specified in the dimension statement of the calling program. (Input)

B — Real band matrix of order N containing a copy of A. (Output)

LDB — Leading dimension of B exactly as specified in the dimension statement of the calling program. (Input)

Algorithm

The routine CRBRB copies the real band matrix A in band storage mode into the real band matrix B in band storage mode.

Example

A real band matrix of order 3, in band storage mode with one upper codiagonal, and one lower codiagonal is copied into another real band matrix also in band storage mode.

CCBCB/DCCBCB (Single/Double precision)

Copy a complex band matrix stored in complex band storage mode.

Usage

CALL CCBCB (N, A, LDA, NLCA, NUCA, B, LDB, NLCB, NUCB)

Arguments

N — Order of the matrices A and B. (Input)

A — Complex band matrix of order N. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement of the calling program. (Input)

B — Complex matrix of order N containing a copy of A. (Output)

LDB — Leading dimension of B exactly as specified in the dimension statement of the calling program. (Input)

Algorithm

The routine CCBCB copies the complex band matrix A in band storage mode into the complex band matrix B in band storage mode.

Example

A complex band matrix of order 3 in band storage mode with one upper codiagonal and one lower codiagonal is copied into another complex band matrix in band storage mode.

Output

CRGRB/DCRGRB (Single/Double precision)

Convert a real general matrix to a matrix in band storage mode.

Usage

CALL CRGRB (N, A, LDA, NLC, NUC, B, LDB)

Arguments

N — Order of the matrices A and B. (Input)

A — Real N by N matrix. (Input)

LDA — Leading dimension of A exactly as specified in the dimension statement of the calling program. (Input)

B — Real ( NUC + 1 + NLC) by N array containing the band matrix in band storage mode. (Output)

LDB — Leading dimension of B exactly as specified in the dimension statement of the calling program. (Input)

Algorithm

The routine CRGRB converts the real general N × N matrix A with mX = NUC upper codiagonals and mO = NLC lower codiagonals into the real band matrix B of order N. The first mX rows of B then contain the upper codiagonals of A, the next row contains the main diagonal of A, and the last mO rows of B contain the lower codiagonals of A.

Example

A real 4 × 4 matrix with one upper codiagonal and three lower codiagonals is copied to a real band matrix of order 4 in band storage mode.