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APPENDIX A

FLAME algorithms for the BLAS-3 routines

The following algorithms correspond to the algorithmic variants of the routines SYMM (Figure A.1), SYR2K (Figure A.2), TRMM (Figure A.3), and TRSM (Figure A.4) developed and evaluated in Chapter 3. Algorithms are described using the FLAME notation.

215

APPENDIX A. FLAME ALGORITHMS FOR THE BLAS-3 ROUTINES

Algorithm: SYMM PP PM(C, A, B)

Partition C →	CB !	, B →	BB !	,
	CT		BT

AT L AT R

A →

ABL ABR

where CT , BT have 0 rows, AT L is 0 × 0 while m(CT ) < m(C) do

Determine block size b Repartition

C1 ,

B1 ,

→ C2

→

AT L

AT R

ABR ! →

ABL

A20

A21

A22

where

C ,

have b rows, A11 is b

Symm

C0 := C0 + A10T B1

(GEMM)

C1 := C1 + A11B1

(SYMM)

C2 := C2 + A21B1

(GEMM)

Symm

C1 := C1 + A10B0

(GEMM)

C1 := C1 + A11B1

(SYMM)

C1 := C1 + A21T B2

(GEMM)

Continue with

B1 ,

C1 ,

← C2

←

AT L AT R

A10

A11

A12

! ←

A00

A01

A02

A20

A21

A22

ABL

ABR

endwhile

Algorithm: SYMM MP(C, A, B)

Partition	C → CL	CR ,
Partition	C → CL	CR ,
	C
B → BL	BR
where	L has 0 columns,

BL has 0 columns while n(CL) < n(C) do

Determine block size b Repartition

CL CR → C0 C1 C2 ,

BL BR → B0 B1 B2 where C1 has b columns, B1 has b columns

C1 := C1 + AB1 (SYMM)

Continue with

CL CR ← C0 C1 C2 ,

endwhile	B		← B		B		B

BL		R		0		1		2

Figure A.1: Algorithms for SYMM.

216

Algorithm: SYR2K MP(A, B, C)

Partition A →	AB !	, B →	BB !	,
	AT		BT

CT L CT R

C →

CBL CBR

where AT has 0 rows, BT has 0 rows, CT L is 0 × 0

while m(AT ) < m(A) do Determine block size b Repartition

A1 ,

B1 ,

! →

→

CT L

CT R

! →

CBL

CBR

C20

C21

C22

has b

where

rows, B1 has b rows,

C11 is b × b

Syr2k

C01 := C01 + A0B1T

(GEMM)

C01 := C01 + B0A1T

(GEMM)

C11 := C11 + A1B1T + B1A1T

(SYR2K)

Syr2k

C12 := C12 + A1B2T

(GEMM)

C12 := C12 + B1A2T

(GEMM)

C11 := C11 + A1B1T + B1A1T

(SYR2K)

Continue with

B1 ,

A1 ,

! ←

←

CT L CT R

C10

C11

C12

! ←

C00

C01

C02

C20

C21

C22

CBL

CBR

endwhile

Algorithm: SYR2K PP(A, B)

Partition

A → AL

AR ,

B → BL

where

L has 0 columns,

BL has 0 columns

while n(AL) < n(A) do

Determine block size b

Repartition

AR → A0

A2 ,

where

BL BR

→ B0

A1 has b columns,

B1 has b columns

C := C + A1B1T + B1A1T

(SYR2K)

Continue with

AR ← A0

A2 ,

endwhile

← B

Figure A.2: Algorithms for SYR2K.

217

APPENDIX A. FLAME ALGORITHMS FOR THE BLAS-3 ROUTINES

Algorithm: TRMM PP PM(A, B)

AT L

AT R

Partition

A →

, B →

ABL

ABR

where

AT L is 0 × 0, BT has 0 rows

while m(AT L) < m(A) do

Determine block size b

Repartition

AT L AT R

A10

A11

A12 ,

! →

A00

A01

A02

A20

A21

A22

ABL

ABR

BB ! →

where

A is b b , B1 has b rows

Trmm

B0 := B0 + A01B1

(GEMM)

B1 := A11B1

(TRMM)

Trmm

B1 := A11B1

(TRMM)

B1 := B1 + A12B2

(GEMM)

Continue with

AT L AT R

A10

A11

A12 ,

! ←

A00

A01

A02

A20

A21

A22

ABL

ABR

BB ! ←

endwhile

Algorithm: TRMM MP(A, B)

where		B
Partition		B →			BL	BR
			L has		0 columns
while n(BL) < n(B)						do
Determine block size b
Repartition

BL	BR			→ B0		B1	B2
where		B1 has b columns

B1 := AB1 (TRMM)

Continue with

BL BR ← B0 B1 B2 endwhile

Figure A.3: Algorithms for TRMM.

218

Algorithm: TRSM PP PM(A, B)

AT L

AT R

Partition

A →

! ,

ABL

ABR

is 0 × 0,

B → BL

where

T L

BL has 0 columns

while m(AT L) < m(A)

Determine block size b

Repartition

AT L AT R

A10

A11

A12 ,

! →

ABL

ABR

A20

A21

A22

where

→ B0

A11 is b

b ,

B1 has b columns

Trsm

B1 := B1 − B0A10T

(GEMM)

B1 := B1A11−T

(TRSM)

Trsm

B1 := B1A11−T

(TRSM)

B2 := B2 − B1A21T

(GEMM)

Continue with

AT L AT R

A10

A11

A12 ,

! ←

A00

A01

A02

A20

A21

A22

ABL

ABR

endwhile

← B

Algorithm: TRSM MP(A, B)

Partition B →

BB where BT has 0 rows

while m(BT ) < m(B) do Determine block size b Repartition

	BT		B1
		!		0
			→	B
	BB

			B2

			has
where			B1		b rows

B1 := B1A−T				(TRSM)

Continue with

BT			B1
	!			B0
		←	B2
BB

endwhile

Figure A.4: Algorithms for TRSM.

219

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