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 (see Specification of the Level 2 BLAS). An additional extended set of operations on matrices is called the Level 3 BLAS (see Specification of the Level 3 BLAS).

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:

 

I	Integer
S	Real	C	Complex
D	Double	Z	Double Complex
SD	Single and Double	CZ	Single and Double Complex
DQ	Double and Quadruple	ZQ	Double and Quadruple Complex

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 and DXYZ refer to a third vector argument z. The storage increment INCZ for z is defined like INCX and 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. 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 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:

 

IMPLICIT INTEGER (I-N)

IMPLICIT REAL S

IMPLICIT DOUBLE PRECISION D

IMPLICIT COMPLEX C

IMPLICIT DOUBLE COMPLEX Z

 

INTEGER IX(MX)

REAL SX(MX), SY(MY), SZ(MZ),

SPARAM(5)

DOUBLE PRECISION DX(MX), DY(MY), DZ(MZ),

DPARAM(5)

DOUBLE PRECISION DACC(2), DZACC(4)

COMPLEX CX(MX), CY(MY)

DOUBLE COMPLEX ZX(MX), ZY(MY)

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 1 – Level 1 Basic Linear Algebra Subprograms

Operation	Integer	Real	Double	Complex	Double-Complex	Subprogram
xi ← a	ISET	SSET	DSET	CSET	ZSET	Set a Vector to a Constant Value
yi ← xi	ICOPY	SCOPY	DCOPY	CCOPY	ZCOPY	Copy a Vector
xi ← axi a ∈ R		SSCAL	DSCAL	CSCAL CSSCAL	ZSCAL ZDSCAL	Scale a Vector
yi ← axi a ∈ R		SVCAL	DVCAL	CVCAL CSVCAL	ZVCAL ZDVCAL	Multiply a Vector by a Constant
xi ← xi + a	IADD	SADD	DADD	CADD	ZADD	Add a Constant to a Vector
xi ← a ‑ xi	ISUB	SSUB	DSUB	CSUB	ZSUB	Subtract a Vector from a Constant
yi ← axi + yi		SAXPY	DAXPY	CAXPY	ZAXPY	Constant Times a Vector Plus a Vector
yi ↔ xi	ISWAP	SSWAP	DSWAP	CSWAP	ZSWAP	Swap Two Vectors
x ⋅ y ⋅ y		SDOT	DDOT	CDOTU CDOTC	ZDOTU ZDOTC	Swap Two Vectors
x ⋅ y † ⋅ y †		DSDOT		CZDOTU CZDOTC	ZQDOTU ZQDOTC	Dot Product with Higher Precision Accumulation
a + x ⋅ y † a + ⋅ y †		SDSDOT	DQDDOT	CZUDOT CZCDOT	ZQUDOT ZQCDOT	Constant Plus Dot Product with Higher Precision Accumulation
b + x ⋅ y † ACC + b + x ⋅ y †		SDDOTI SDDOTA	DQDOTI DQDOTA	CZDOTI CZDOTA	ZQDOTI ZQDOTA	Dot Product Using the Accumulator
zi ← xiyi		SHPROD	DHPROD			Hadamard Product
∑xiyizi		SXYZ	DXYZ			Triple Inner Product
∑xi	ISUM	SSUM	DSUM			Sum of the Elements of a Vector
∑∣xi∣		SASUM	DASUM	SCASUM	DZASUM	Sum of the Absolute Values of the Elements of a Vector
∥x∥2		SNRM2	DNRM2	SCNRM2	DZNRM2	Euclidean or ℓ2 Norm of a Vector
Πxi		SPRDCT	DPRDCT			Product of the Elements of a Vector
i : xi = minj xj	IIMIN	ISMIN	IDMIN			Index of Element Having Minimum Value
i : xi = maxj xj	IIMAX	ISMAX	IDMAX			Index of Element Having Maximum Value
i : ∣xi∣ = minj ∣xj∣		ISAMIN	IDAMIN	ICAMIN	IZAMIN	Index of Element Having Minimum Absolute Value
i : ∣ xi∣ = maxj ∣xj∣		ISAMAX	IDAMAX	ICAMAX	IZAMAX	Index of Element Having Maximum Absolute Value
Construct Givens rotation		SROTG	DROTG	CROTG	ZROTG	Construct a Givens Plane Rotation
Apply Givens rotation		SROT	DROT	CROT CSROT	ZROT ZDROT	Apply a Plane Rotation
Construct modified Givens transform		SROTMG	DROTMG			Construct a Modified Givens Transformation
Apply modified Givens transform		SROTM	DROTM	CSROTM	ZDROTM	Apply a Modified Givens Transformation

†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 xi ← a for i = 1, 2, …, 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 yi ← xi for i = 1, 2, …, 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 xi ← axi for i = 1, 2, …, 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 yi ← axi for i = 1, 2, …, 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 xi ← xi + a for i = 1, 2, …, 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 xi ← a ‑ xi for i = 1, 2, …, 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 yi ← axi + yi for i = 1, 2, …, 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 yi ↔ xi for i = 1, 2, …, 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

 

The function subprograms CDOTC and ZDOTC compute

 

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

Dot Product with Higher Precision Accumulation

 

*DW = DSDOT (N, SX, INCX, SY, INCY)

CW = CZDOTC (N, CX, INCX, CY, INCY)

CW = CZDOTU (N, CX, INCX, CY, INCY)

ZW = ZQDOTC (N, ZX, INCX, ZY, INCY)

ZW = ZQDOTU (N, ZX, INCX, ZY, INCY)

The function subprogram DSDOT computes

 

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

 

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

 

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

 

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

 

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

Dot Product Using the Accumulator

 

SW = SDDOTI (N, SB, DACC, SX, INCX, SY, INCY)

SW = SDDOTA (N, SB, DACC, SX, INCX, SY, INCY)

CW = CZDOTI (N, CB, DACC, CX, INCX, CY, INCY)

CW = CZDOTA (N, CB, DACC, CX, INCX, CY, INCY)

*DW = DQDOTI (N, DB, DACC, DX, INCX, DY, INCY)

*DW = DQDOTA (N, DB, DACC, DX, INCX, DY, INCY)

ZW = ZQDOTI (N, ZB, DZACC, ZX, INCX, ZY, INCY)

ZW = ZQDOTA (N, ZB, DZACC, ZX, INCX, ZY, INCY)

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

 

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 zi ← xiyi for i = 1, 2, …, 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

 

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

 

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

 

The function subprograms SCASUM and DZASUM compute

 

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

Euclidean or ℓ2 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

 

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

 

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 xi = min1 ≤ j ≤ Nxj. 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 xi = max1 ≤ j ≤ Nxj. 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 ∣xi∣ = min1 ≤ j ≤ N∣xj∣. The function subprograms ICAMIN and IZAMIN compute the smallest index i such that

 

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
∣xi∣ = max1 ≤ j≤N∣xj∣. The function subprograms ICAMAX and IZAMAX compute the smallest index i such that

 

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

 

and

 

where r = σ(a2 + b2)1/2 and

 

Then, the values c, s and r satisfy the matrix equation

 

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

 

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

 

If ∣z∣ > 1, then set

 

Apply a Plane Rotation

 

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

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

CALL CSROT (N, CX, INCX, CY, INCY, SC, SS)

CALL ZDROT (N, ZX, INCX, ZY, INCY, DC, DS)

These subprograms compute

 

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]T 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

 

respectively. That is,

 

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,

 

and

 

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

PARAM(1) = 0. In this case,

 

and

 

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

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

 

PARAM(2) = ‑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 x1are 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 PARAM(1) = 1.0, then these subprograms compute

 

If PARAM(1) = 0.0, then the subprograms compute

 

If PARAM(1) = 1.0, then the subprograms compute

 

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 Reference Material. 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:

 

S   Real	C    Complex
D   Double	Z   Double Complex

The root names for the kind of matrix:

 

GE   General	GB   General Band	SP   Symmetric Packed
SY   Symmetric	SB   Symmetric Band	TP   Triangular Packed
HE   Hermitian	HB   Hermitian Band	HP   Hermitian Packed
TR   Triangular	TB   Triangular Band

The suffixes for the type of operation:

 

MV	Matrix-Vector Product	SV	Solve for Vector
R	Rank-One Update
RU	Rank-One Update, Unconjugated	RC	Rank-One Update, Conjugated
R2	Rank-Two Update
MM	Matrix-Multiply	SM	Symmetric Matrix Multiply
RK	Rank-K Update	R2K	Rank 2K Update

1IMSL does not support 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:

 

TRANS, TRANSA, TRANSB	'N'	No Transpose
	'T'	Transpose
	'C'	Conjugate and Transpose
UPLO	'L'	Lower Triangular
	'U'	Upper Triangular
DIAGNL	'N'	Non-unit Triangular
	'U'	Unit Triangular
SIDE	'L'	Multiply “A” Matrix on Left side, or
	'R'	Right side of the “B” matrix

Note: See the “Triangular Mode” section in the Reference Material 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 βy or βC do not require that the contents of y or C be defined when β = 0. In that case, the expression βy or βC 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

 

INTEGER INCX, INCY, NCODA, NLCA, NUCA, LDA, M, N

CHARACTER*1 DIAGNL, TRANS, UPLO

 

REAL SALPHA, SBETA, SX(*), SY(*), SA(LDA,*)

DOUBLE PRECISION DALPHA, DBETA, DX(*), DY(*), DA(LDA,*)

COMPLEX CALPHA, CBETA, CX(*), CY(*), CA(LDA,*)

DOUBLE COMPLEX ZALPHA, ZBETA, ZX(*), ZY(*), ZA(LDA,*)

There is a lower bound on the leading dimension LDA. It must be ≥ 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

 

INTEGER K, LDA, LDB, LDC, M, N

CHARACTER*1 DIAGNL, TRANS, TRANSA, TRANSB, SIDE, UPLO

REAL SALPHA, SBETA, SA(LDA,*), SB(LDB,*), SC(LDC,*)

DOUBLE PRECISION DALPHA, DBETA, DA(LDA,*), DB(LDB,*), DC(LDC,*)

COMPLEX CALPHA, CBETA, CA(LDA,*), CB(LDB,*), CC(LDC,*)

DOUBLE COMPLEX ZALPHA, ZBETA, ZA(LDA,*), ZB(LDB,*), ZC(LDC,*)

Each of the integers K, M, N must be ≥ 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 ≥ the number of rows in the matrix that is contained in this array.

Table 2 describes the Level 2 and 3 BLAS subprograms.

Table 2 – Level 2 and Level 3 Basic Linear Algebra Subprograms –

Operation	Real	Double	Complex	Double-Complex
Matrix–Vector Multiply, General	SGEMV	DGEMV	CGEMV	ZGEMV
Matrix–Vector Multiply, Banded	SGBMV	DGBMV	CGBMV	ZGBMV
Matrix-Vector Multiply, Hermitian			CHEMV	ZHEMV
Matrix-Vector Multiply, Packed Hermitian			CHPMV	ZHPMV
Matrix-Vector Multiply, Hermitian and Banded			CHBMV	ZHBMV
Matrix-Vector Multiply, Symmetric and Real	SSYMV	DSYMV
Matrix-Vector Multiply, Packed Symmetric and Real	SSPMV	DSPMV
Matrix-Vector Multiply, Symmetric and Banded	SSBMV	DSBMV
Matrix-Vector Multiply, Triangular	STRMV	DTRMV	CTRMV	ZTRMV
Matrix-Vector Multiply, Packed Triangular	STPMV	DTPMV	CTPMV	ZTPMV
Matrix-Vector Multiply, Triangular and Banded	STBMV	DTBMV	CTBMV	ZTBMV
Matrix-Vector Solve, Triangular	STRSV	DTRSV	CTRSV	ZTRSV
Matrix-Vector Solve, Triangular and Banded	STBSV	DTBSV	CTBSV	ZTBSV
Matrix-Vector Solve, Packed Triangular	STPSV	DTPSV	CTPSV	ZTPSV
Rank-One Matrix Update, General and Real	SGER	DGER
Rank-One Matrix Update, General, Complex, and Transpose			CGERU	ZGERU
Rank-One Matrix Update, General, Complex, and Conjugate Transpose			CGERC	ZGERC
Rank-One Matrix Update, Hermitian and Conjugate Transpose			CHER	ZHER
Rank-Two Matrix Update, Hermitian and Conjugate Transpose			CHER2	ZHER2
Rank-Two Matrix Update, Packed Hermitian and Conjugate Transpose			CHPR2	ZHPR2
Rank-One Matrix Update, Symmetric and Real	SSYR	DSYR
Rank-One Matrix Update, Packed Symmetric and Real	SSPR	DSPR
Rank-One Matrix Update, Packed Hermitian			CHPR	ZHPR
Rank-Two Matrix Update, Symmetric and Real	SSYR2	DSYR2
Rank-Two Matrix Update, Packed Symmetric and Real	SSPR2	DSPR2
Matrix-Matrix Multiply, General	SGEMM	DGEMM	CGEMM	ZGEMM
Matrix-Matrix Multiply, Symmetric	SSYMM	DSYMM	CSYMM	ZSYMM
Matrix-Matrix Multiply, Hermitian			CHEMM	ZHEMM
Rank-k Update, Symmetric	SSYRK	DSYRK	CSYRK	ZSYRK
Rank-k Update, Hermitian			CHERK	ZHERK
Rank-2k, Symmetric	SSYR2K	DSYR2K	CSYR2K	ZSYR2K
Rank-2k, Hermitian			CHER2K	ZHER2K
Matrix-Matrix Multiply, Triangular	STRMM	DTRMM	CTRMM	ZTRMM
Matrix-Matrix Solve, Triangular	STRSM	DTRSM	CTRSM	ZTRSM

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 ← αATx + βy, or for complex data,

 

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, BETA, 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 ← αATx + βy, or for complex data,

 

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 complex 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 the upper or lower triangular part. The character flag UPLO determines the part used.

Matrix-Vector Multiply, Packed Hermitian

 

CALL CHPMV (UPLO, N, CALPHA, CAP, CX, INCX, CBETA, CY, INCY)

CALL ZHPMV (UPLO, N, ZALPHA, ZAP, ZX, INCX, ZBETA, ZY, INCY)

For complex 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 the packed 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 the upper or lower triangular part. The character flag UPLO determines the part used.

Matrix-Vector Multiply, Packed Symmetric and Real

 

CALL SSPMV (UPLO, N, SALPHA, SAP, SX, INCX, SBETA, SY, INCY)

CALL DSPMV (UPLO, N, DALPHA, DAP, 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 packed triangular matrix. The matrix A is either referenced using the packed 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, DIAGNL, N, SA, LDA, SX, INCX)

CALL DTRMV (UPLO, TRANS, DIAGNL, N, DA, LDA, DX, INCX)

CALL CTRMV (UPLO, TRANS, DIAGNL, N, CA, LDA, CX, INCX)

CALL ZTRMV (UPLO, TRANS, DIAGNL, 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 ← ATx, or for complex data,

 

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 DIAGNL determine the part of the matrix used and the operation performed.

Matrix-Vector Multiply, Packed Triangular

 

CALL STPMV (UPLO, TRANS, DIAGNL, N, SAP, SX, INCX)

CALL DTPMV (UPLO, TRANS, DIAGNL, N, DAP, DX, INCX)

CALL CTPMV (UPLO, TRANS, DIAGNL, N, CAP, CX, INCX)

CALL ZTPMV (UPLO, TRANS, DIAGNL, N, ZAP, ZX, INCX)

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

 

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

Matrix-Vector Multiply, Triangular and Banded

 

CALL STBMV (UPLO, TRANS, DIAGNL, N, NCODA, SA, LDA, SX, INCX)

CALL DTBMV (UPLO, TRANS, DIAGNL, N, NCODA, DA, LDA, DX, INCX)

CALL CTBMV (UPLO, TRANS, DIAGNL, N, NCODA, CA, LDA, CX, INCX)

CALL ZTBMV (UPLO, TRANS, DIAGNL, N, NCODA, ZA, LDA, ZX, INCX)

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 ← ATx, or for complex data,

 

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 DIAGNL determine the part of the matrix used and the operation performed.

Matrix-Vector Solve, Triangular

 

CALL STRSV (UPLO, TRANS, DIAGNL, N, SA, LDA, SX, INCX)

CALL DTRSV (UPLO, TRANS, DIAGNL, N, DA, LDA, DX, INCX)

CALL CTRSV (UPLO, TRANS, DIAGNL, N, CA, LDA, CX, INCX)

CALL ZTRSV (UPLO, TRANS, DIAGNL, 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-1x, x ← (A-1)Tx, or for complex data,

 

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 DIAGNL determine the part of the matrix used and the operation performed.

Matrix-Vector Solve, Triangular and Banded

 

CALL STBSV (UPLO, TRANS, DIAGNL, N, NCODA, SA, LDA, SX, INCX)

CALL DTBSV (UPLO, TRANS, DIAGNL, N, NCODA, DA, LDA, DX, INCX)

CALL CTBSV (UPLO, TRANS, DIAGNL, N, NCODA, CA, LDA, CX, INCX)

CALL ZTBSV (UPLO, TRANS, DIAGNL, 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-1x, x ← (A-T) -1x, or for complex data,

 

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 DIAGNL determine the part of the matrix used and the operation performed.

Matrix-Vector Solve, Packed Triangular

 

CALL STPSV (UPLO, TRANS, DIAGNL, N, SAP, SX, INCX)

CALL DTPSV (UPLO, TRANS, DIAGNL, N, DAP, DX, INCX)

CALL CTPSV (UPLO, TRANS, DIAGNL, N, CAP, CX, INCX)

CALL ZTPSV (UPLO, TRANS, DIAGNL, N, ZAP, ZX, INCX)

For all data types, A is an N × N packed triangular matrix. These subprograms solve x for one of the expressions: x ← A-1x, x ← (A-1)Tx, or for complex data,

 

The matrix A is either referenced using its packed upper or lower triangular part and is unit or nonunit triangular. The character flags UPLO, TRANS, and DIAGNL 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 + αxyT.

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 + αxyT.

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

 

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

 

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-One Matrix Update, Packed Hermitian and Conjugate Transpose

 

CALL CHPR (UPLO, N, SALPHA, CX, INCX, CAP)

CALL ZHPR (UPLO, N, DALPHA, ZX, INCX, ZAP)

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

 

where A is packed 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

 

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-Two Matrix Update, Packed Hermitian and Conjugate Transpose

 

CALL CHPR2 (UPLO, N, CALPHA, CX, INCX, CY, INCY, CAP)

CALL ZHPR2 (UPLO, N, ZALPHA, ZX, INCX, ZY, INCY, ZAP)

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

 

where A is a packed 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 + αxxT 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-One Matrix Update, Packed Symmetric and Real

 

CALL SSPR (UPLO, N, SALPHA, SX, INCX, SAP)

CALL DSPR (UPLO, N, DALPHA, DX, INCX, DAP)

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

Rank-One Matrix Update, Packed Hermitian

 

CALL CHPR (UPLO, N, SALPHA, CX, INCX, CAP)

CALL ZHPR (UPLO, N, DALPHA, ZX, INCX, ZAP)

For all data types, A is an N × N matrix. These subprograms set A ← A + αxxT where A is a packed Hermitian matrix. The matrix A is either referenced using the packed 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 + αxyT + αyxT, 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.

Rank-Two Matrix Update, Packed Symmetric and Real

 

CALL SSPR2 (UPLO, N, SALPHA, SX, INCX, SY, INCY, SAP)

CALL DSPR2 (UPLO, N, DALPHA, DX, INCX, DY, INCY, DAP)

For all data types, A is an N × N matrix. These subprograms set A ← A + αxyT + αyxT, where A is a packed 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 CM×N to one of the expressions:

 

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 CM×N to one of the expressions: C ← αAB + 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 CM×N to one of the expressions: C ← αAB + 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 CM×N to one of the expressions: C ← αAAT + βC or C ← αATA + 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 CN× N to one of the expressions:

 

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, 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 CN× N to one of the expressions:

 

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, 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 CN× N to one of the expressions:

 

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, DIAGNL, M, N, SALPHA, SA, LDA, SB, LDB)

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

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

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

For all data types, these subprograms set BM×N to one of the_expressions:

 

or for complex data,

 

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 DIAGNL determine the part of the matrix used and the operation performed.

Matrix-Matrix Solve, Triangular

 

CALL STRSM (SIDE, UPLO, TRANSA, DIAGNL, M, N, SALPHA, SA, LDA, SB, LDB)

CALL DTRSM (SIDE, UPLO, TRANSA, DIAGNL, M, N, DALPHA, DA, LDA, DB, LDB)

CALL CTRSM (SIDE, UPLO, TRANSA, DIAGNL, M, N, CALPHA, CA, LDA, CB, LDB)

CALL ZTRSM (SIDE, UPLO, TRANSA, DIAGNL, M, N, ZALPHA, ZA, LDA, ZB, LDB)

For all data types, these subprograms set BM×N to one of the expressions:

 

or for complex data,

 

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 DIAGNL determine the part of the matrix used and the operation performed.