* * This file contains the single precision fortran BLAS routines needed * by the Templates. Therefore, if you can link to a BLAS library, this * file is not needed. * subroutine saxpy(n,sa,sx,incx,sy,incy) c c constant times a vector plus a vector. c uses unrolled loop for increments equal to one. c jack dongarra, linpack, 3/11/78. c real sx(1),sy(1),sa integer i,incx,incy,ix,iy,m,mp1,n c if(n.le.0)return if (sa .eq. 0.0) return if(incx.eq.1.and.incy.eq.1)go to 20 c c code for unequal increments or equal increments c not equal to 1 c ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n sy(iy) = sy(iy) + sa*sx(ix) ix = ix + incx iy = iy + incy 10 continue return c c code for both increments equal to 1 c c c clean-up loop c 20 m = mod(n,4) if( m .eq. 0 ) go to 40 do 30 i = 1,m sy(i) = sy(i) + sa*sx(i) 30 continue if( n .lt. 4 ) return 40 mp1 = m + 1 do 50 i = mp1,n,4 sy(i) = sy(i) + sa*sx(i) sy(i + 1) = sy(i + 1) + sa*sx(i + 1) sy(i + 2) = sy(i + 2) + sa*sx(i + 2) sy(i + 3) = sy(i + 3) + sa*sx(i + 3) 50 continue return end subroutine scopy(n,sx,incx,sy,incy) c c copies a vector, x, to a vector, y. c uses unrolled loops for increments equal to 1. c jack dongarra, linpack, 3/11/78. c real sx(1),sy(1) integer i,incx,incy,ix,iy,m,mp1,n c if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 c c code for unequal increments or equal increments c not equal to 1 c ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n sy(iy) = sx(ix) ix = ix + incx iy = iy + incy 10 continue return c c code for both increments equal to 1 c c c clean-up loop c 20 m = mod(n,7) if( m .eq. 0 ) go to 40 do 30 i = 1,m sy(i) = sx(i) 30 continue if( n .lt. 7 ) return 40 mp1 = m + 1 do 50 i = mp1,n,7 sy(i) = sx(i) sy(i + 1) = sx(i + 1) sy(i + 2) = sx(i + 2) sy(i + 3) = sx(i + 3) sy(i + 4) = sx(i + 4) sy(i + 5) = sx(i + 5) sy(i + 6) = sx(i + 6) 50 continue return end real function sdot(n,sx,incx,sy,incy) c c forms the dot product of two vectors. c uses unrolled loops for increments equal to one. c jack dongarra, linpack, 3/11/78. c real sx(1),sy(1),stemp integer i,incx,incy,ix,iy,m,mp1,n c stemp = 0.0e0 sdot = 0.0e0 if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 c c code for unequal increments or equal increments c not equal to 1 c ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n stemp = stemp + sx(ix)*sy(iy) ix = ix + incx iy = iy + incy 10 continue sdot = stemp return c c code for both increments equal to 1 c c c clean-up loop c 20 m = mod(n,5) if( m .eq. 0 ) go to 40 do 30 i = 1,m stemp = stemp + sx(i)*sy(i) 30 continue if( n .lt. 5 ) go to 60 40 mp1 = m + 1 do 50 i = mp1,n,5 stemp = stemp + sx(i)*sy(i) + sx(i + 1)*sy(i + 1) + * sx(i + 2)*sy(i + 2) + sx(i + 3)*sy(i + 3) + sx(i + 4)*sy(i + 4) 50 continue 60 sdot = stemp return end * SUBROUTINE SGEMM ( TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, \$ BETA, C, LDC ) * .. Scalar Arguments .. CHARACTER*1 TRANSA, TRANSB INTEGER M, N, K, LDA, LDB, LDC REAL ALPHA, BETA * .. Array Arguments .. REAL A( LDA, * ), B( LDB, * ), C( LDC, * ) * .. * * Purpose * ======= * * SGEMM performs one of the matrix-matrix operations * * 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 by k matrix, op( B ) a k by n matrix and C an m by n matrix. * * Parameters * ========== * * TRANSA - CHARACTER*1. * On entry, TRANSA specifies the form of op( A ) to be used in * the matrix multiplication as follows: * * TRANSA = 'N' or 'n', op( A ) = A. * * TRANSA = 'T' or 't', op( A ) = A'. * * TRANSA = 'C' or 'c', op( A ) = A'. * * Unchanged on exit. * * TRANSB - CHARACTER*1. * On entry, TRANSB specifies the form of op( B ) to be used in * the matrix multiplication as follows: * * TRANSB = 'N' or 'n', op( B ) = B. * * TRANSB = 'T' or 't', op( B ) = B'. * * TRANSB = 'C' or 'c', op( B ) = B'. * * Unchanged on exit. * * M - INTEGER. * On entry, M specifies the number of rows of the matrix * op( A ) and of the matrix C. M must be at least zero. * Unchanged on exit. * * N - INTEGER. * On entry, N specifies the number of columns of the matrix * op( B ) and the number of columns of the matrix C. N must be * at least zero. * Unchanged on exit. * * K - INTEGER. * On entry, K specifies the number of columns of the matrix * op( A ) and the number of rows of the matrix op( B ). K must * be at least zero. * Unchanged on exit. * * ALPHA - REAL . * On entry, ALPHA specifies the scalar alpha. * Unchanged on exit. * * A - REAL array of DIMENSION ( LDA, ka ), where ka is * k when TRANSA = 'N' or 'n', and is m otherwise. * Before entry with TRANSA = 'N' or 'n', the leading m by k * part of the array A must contain the matrix A, otherwise * the leading k by m part of the array A must contain the * matrix A. * Unchanged on exit. * * LDA - INTEGER. * On entry, LDA specifies the first dimension of A as declared * in the calling (sub) program. When TRANSA = 'N' or 'n' then * LDA must be at least max( 1, m ), otherwise LDA must be at * least max( 1, k ). * Unchanged on exit. * * B - REAL array of DIMENSION ( LDB, kb ), where kb is * n when TRANSB = 'N' or 'n', and is k otherwise. * Before entry with TRANSB = 'N' or 'n', the leading k by n * part of the array B must contain the matrix B, otherwise * the leading n by k part of the array B must contain the * matrix B. * Unchanged on exit. * * LDB - INTEGER. * On entry, LDB specifies the first dimension of B as declared * in the calling (sub) program. When TRANSB = 'N' or 'n' then * LDB must be at least max( 1, k ), otherwise LDB must be at * least max( 1, n ). * Unchanged on exit. * * BETA - REAL . * On entry, BETA specifies the scalar beta. When BETA is * supplied as zero then C need not be set on input. * Unchanged on exit. * * C - REAL array of DIMENSION ( LDC, n ). * Before entry, the leading m by n part of the array C must * contain the matrix C, except when beta is zero, in which * case C need not be set on entry. * On exit, the array C is overwritten by the m by n matrix * ( alpha*op( A )*op( B ) + beta*C ). * * LDC - INTEGER. * On entry, LDC specifies the first dimension of C as declared * in the calling (sub) program. LDC must be at least * max( 1, m ). * Unchanged on exit. * * * Level 3 Blas routine. * * -- Written on 8-February-1989. * Jack Dongarra, Argonne National Laboratory. * Iain Duff, AERE Harwell. * Jeremy Du Croz, Numerical Algorithms Group Ltd. * Sven Hammarling, Numerical Algorithms Group Ltd. * * * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. External Subroutines .. EXTERNAL XERBLA * .. Intrinsic Functions .. INTRINSIC MAX * .. Local Scalars .. LOGICAL NOTA, NOTB INTEGER I, INFO, J, L, NCOLA, NROWA, NROWB REAL TEMP * .. Parameters .. REAL ONE , ZERO PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) * .. * .. Executable Statements .. * * Set NOTA and NOTB as true if A and B respectively are not * transposed and set NROWA, NCOLA and NROWB as the number of rows * and columns of A and the number of rows of B respectively. * NOTA = LSAME( TRANSA, 'N' ) NOTB = LSAME( TRANSB, 'N' ) IF( NOTA )THEN NROWA = M NCOLA = K ELSE NROWA = K NCOLA = M END IF IF( NOTB )THEN NROWB = K ELSE NROWB = N END IF * * Test the input parameters. * INFO = 0 IF( ( .NOT.NOTA ).AND. \$ ( .NOT.LSAME( TRANSA, 'C' ) ).AND. \$ ( .NOT.LSAME( TRANSA, 'T' ) ) )THEN INFO = 1 ELSE IF( ( .NOT.NOTB ).AND. \$ ( .NOT.LSAME( TRANSB, 'C' ) ).AND. \$ ( .NOT.LSAME( TRANSB, 'T' ) ) )THEN INFO = 2 ELSE IF( M .LT.0 )THEN INFO = 3 ELSE IF( N .LT.0 )THEN INFO = 4 ELSE IF( K .LT.0 )THEN INFO = 5 ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN INFO = 8 ELSE IF( LDB.LT.MAX( 1, NROWB ) )THEN INFO = 10 ELSE IF( LDC.LT.MAX( 1, M ) )THEN INFO = 13 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'SGEMM ', INFO ) RETURN END IF * * Quick return if possible. * IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR. \$ ( ( ( ALPHA.EQ.ZERO ).OR.( K.EQ.0 ) ).AND.( BETA.EQ.ONE ) ) ) \$ RETURN * * And if alpha.eq.zero. * IF( ALPHA.EQ.ZERO )THEN IF( BETA.EQ.ZERO )THEN DO 20, J = 1, N DO 10, I = 1, M C( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ELSE DO 40, J = 1, N DO 30, I = 1, M C( I, J ) = BETA*C( I, J ) 30 CONTINUE 40 CONTINUE END IF RETURN END IF * * Start the operations. * IF( NOTB )THEN IF( NOTA )THEN * * Form C := alpha*A*B + beta*C. * DO 90, J = 1, N IF( BETA.EQ.ZERO )THEN DO 50, I = 1, M C( I, J ) = ZERO 50 CONTINUE ELSE IF( BETA.NE.ONE )THEN DO 60, I = 1, M C( I, J ) = BETA*C( I, J ) 60 CONTINUE END IF DO 80, L = 1, K IF( B( L, J ).NE.ZERO )THEN TEMP = ALPHA*B( L, J ) DO 70, I = 1, M C( I, J ) = C( I, J ) + TEMP*A( I, L ) 70 CONTINUE END IF 80 CONTINUE 90 CONTINUE ELSE * * Form C := alpha*A'*B + beta*C * DO 120, J = 1, N DO 110, I = 1, M TEMP = ZERO DO 100, L = 1, K TEMP = TEMP + A( L, I )*B( L, J ) 100 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = ALPHA*TEMP ELSE C( I, J ) = ALPHA*TEMP + BETA*C( I, J ) END IF 110 CONTINUE 120 CONTINUE END IF ELSE IF( NOTA )THEN * * Form C := alpha*A*B' + beta*C * DO 170, J = 1, N IF( BETA.EQ.ZERO )THEN DO 130, I = 1, M C( I, J ) = ZERO 130 CONTINUE ELSE IF( BETA.NE.ONE )THEN DO 140, I = 1, M C( I, J ) = BETA*C( I, J ) 140 CONTINUE END IF DO 160, L = 1, K IF( B( J, L ).NE.ZERO )THEN TEMP = ALPHA*B( J, L ) DO 150, I = 1, M C( I, J ) = C( I, J ) + TEMP*A( I, L ) 150 CONTINUE END IF 160 CONTINUE 170 CONTINUE ELSE * * Form C := alpha*A'*B' + beta*C * DO 200, J = 1, N DO 190, I = 1, M TEMP = ZERO DO 180, L = 1, K TEMP = TEMP + A( L, I )*B( J, L ) 180 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = ALPHA*TEMP ELSE C( I, J ) = ALPHA*TEMP + BETA*C( I, J ) END IF 190 CONTINUE 200 CONTINUE END IF END IF * RETURN * * End of SGEMM . * END SUBROUTINE SGEMV ( TRANS, M, N, ALPHA, A, LDA, X, INCX, \$ BETA, Y, INCY ) * .. Scalar Arguments .. REAL ALPHA, BETA INTEGER INCX, INCY, LDA, M, N CHARACTER*1 TRANS * .. Array Arguments .. REAL A( LDA, * ), X( * ), Y( * ) * .. * * Purpose * ======= * * SGEMV performs one of the matrix-vector operations * * y := alpha*A*x + beta*y, or y := alpha*A'*x + beta*y, * * where alpha and beta are scalars, x and y are vectors and A is an * m by n matrix. * * Parameters * ========== * * TRANS - CHARACTER*1. * On entry, TRANS specifies the operation to be performed as * follows: * * TRANS = 'N' or 'n' y := alpha*A*x + beta*y. * * TRANS = 'T' or 't' y := alpha*A'*x + beta*y. * * TRANS = 'C' or 'c' y := alpha*A'*x + beta*y. * * Unchanged on exit. * * M - INTEGER. * On entry, M specifies the number of rows of the matrix A. * M must be at least zero. * Unchanged on exit. * * N - INTEGER. * On entry, N specifies the number of columns of the matrix A. * N must be at least zero. * Unchanged on exit. * * ALPHA - REAL . * On entry, ALPHA specifies the scalar alpha. * Unchanged on exit. * * A - REAL array of DIMENSION ( LDA, n ). * Before entry, the leading m by n part of the array A must * contain the matrix of coefficients. * Unchanged on exit. * * LDA - INTEGER. * On entry, LDA specifies the first dimension of A as declared * in the calling (sub) program. LDA must be at least * max( 1, m ). * Unchanged on exit. * * X - REAL array of DIMENSION at least * ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n' * and at least * ( 1 + ( m - 1 )*abs( INCX ) ) otherwise. * Before entry, the incremented array X must contain the * vector x. * Unchanged on exit. * * INCX - INTEGER. * On entry, INCX specifies the increment for the elements of * X. INCX must not be zero. * Unchanged on exit. * * BETA - REAL . * On entry, BETA specifies the scalar beta. When BETA is * supplied as zero then Y need not be set on input. * Unchanged on exit. * * Y - REAL array of DIMENSION at least * ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n' * and at least * ( 1 + ( n - 1 )*abs( INCY ) ) otherwise. * Before entry with BETA non-zero, the incremented array Y * must contain the vector y. On exit, Y is overwritten by the * updated vector y. * * INCY - INTEGER. * On entry, INCY specifies the increment for the elements of * Y. INCY must not be zero. * Unchanged on exit. * * * Level 2 Blas routine. * * -- Written on 22-October-1986. * Jack Dongarra, Argonne National Lab. * Jeremy Du Croz, Nag Central Office. * Sven Hammarling, Nag Central Office. * Richard Hanson, Sandia National Labs. * * * .. Parameters .. REAL ONE , ZERO PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) * .. Local Scalars .. REAL TEMP INTEGER I, INFO, IX, IY, J, JX, JY, KX, KY, LENX, LENY * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. External Subroutines .. EXTERNAL XERBLA * .. Intrinsic Functions .. INTRINSIC MAX * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 IF ( .NOT.LSAME( TRANS, 'N' ).AND. \$ .NOT.LSAME( TRANS, 'T' ).AND. \$ .NOT.LSAME( TRANS, 'C' ) )THEN INFO = 1 ELSE IF( M.LT.0 )THEN INFO = 2 ELSE IF( N.LT.0 )THEN INFO = 3 ELSE IF( LDA.LT.MAX( 1, M ) )THEN INFO = 6 ELSE IF( INCX.EQ.0 )THEN INFO = 8 ELSE IF( INCY.EQ.0 )THEN INFO = 11 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'SGEMV ', INFO ) RETURN END IF * * Quick return if possible. * IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR. \$ ( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) ) \$ RETURN * * Set LENX and LENY, the lengths of the vectors x and y, and set * up the start points in X and Y. * IF( LSAME( TRANS, 'N' ) )THEN LENX = N LENY = M ELSE LENX = M LENY = N END IF IF( INCX.GT.0 )THEN KX = 1 ELSE KX = 1 - ( LENX - 1 )*INCX END IF IF( INCY.GT.0 )THEN KY = 1 ELSE KY = 1 - ( LENY - 1 )*INCY END IF * * Start the operations. In this version the elements of A are * accessed sequentially with one pass through A. * * First form y := beta*y. * IF( BETA.NE.ONE )THEN IF( INCY.EQ.1 )THEN IF( BETA.EQ.ZERO )THEN DO 10, I = 1, LENY Y( I ) = ZERO 10 CONTINUE ELSE DO 20, I = 1, LENY Y( I ) = BETA*Y( I ) 20 CONTINUE END IF ELSE IY = KY IF( BETA.EQ.ZERO )THEN DO 30, I = 1, LENY Y( IY ) = ZERO IY = IY + INCY 30 CONTINUE ELSE DO 40, I = 1, LENY Y( IY ) = BETA*Y( IY ) IY = IY + INCY 40 CONTINUE END IF END IF END IF IF( ALPHA.EQ.ZERO ) \$ RETURN IF( LSAME( TRANS, 'N' ) )THEN * * Form y := alpha*A*x + y. * JX = KX IF( INCY.EQ.1 )THEN DO 60, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = ALPHA*X( JX ) DO 50, I = 1, M Y( I ) = Y( I ) + TEMP*A( I, J ) 50 CONTINUE END IF JX = JX + INCX 60 CONTINUE ELSE DO 80, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = ALPHA*X( JX ) IY = KY DO 70, I = 1, M Y( IY ) = Y( IY ) + TEMP*A( I, J ) IY = IY + INCY 70 CONTINUE END IF JX = JX + INCX 80 CONTINUE END IF ELSE * * Form y := alpha*A'*x + y. * JY = KY IF( INCX.EQ.1 )THEN DO 100, J = 1, N TEMP = ZERO DO 90, I = 1, M TEMP = TEMP + A( I, J )*X( I ) 90 CONTINUE Y( JY ) = Y( JY ) + ALPHA*TEMP JY = JY + INCY 100 CONTINUE ELSE DO 120, J = 1, N TEMP = ZERO IX = KX DO 110, I = 1, M TEMP = TEMP + A( I, J )*X( IX ) IX = IX + INCX 110 CONTINUE Y( JY ) = Y( JY ) + ALPHA*TEMP JY = JY + INCY 120 CONTINUE END IF END IF * RETURN * * End of SGEMV . * END * ============================================================== * SUBROUTINE SGER ( M, N, ALPHA, X, INCX, Y, INCY, A, LDA ) * .. Scalar Arguments .. REAL ALPHA INTEGER INCX, INCY, LDA, M, N * .. Array Arguments .. REAL A( LDA, * ), X( * ), Y( * ) * .. * * Purpose * ======= * * SGER performs the rank 1 operation * * A := alpha*x*y' + A, * * where alpha is a scalar, x is an m element vector, y is an n element * vector and A is an m by n matrix. * * Parameters * ========== * * M - INTEGER. * On entry, M specifies the number of rows of the matrix A. * M must be at least zero. * Unchanged on exit. * * N - INTEGER. * On entry, N specifies the number of columns of the matrix A. * N must be at least zero. * Unchanged on exit. * * ALPHA - REAL . * On entry, ALPHA specifies the scalar alpha. * Unchanged on exit. * * X - REAL array of dimension at least * ( 1 + ( m - 1 )*abs( INCX ) ). * Before entry, the incremented array X must contain the m * element vector x. * Unchanged on exit. * * INCX - INTEGER. * On entry, INCX specifies the increment for the elements of * X. INCX must not be zero. * Unchanged on exit. * * Y - REAL array of dimension at least * ( 1 + ( n - 1 )*abs( INCY ) ). * Before entry, the incremented array Y must contain the n * element vector y. * Unchanged on exit. * * INCY - INTEGER. * On entry, INCY specifies the increment for the elements of * Y. INCY must not be zero. * Unchanged on exit. * * A - REAL array of DIMENSION ( LDA, n ). * Before entry, the leading m by n part of the array A must * contain the matrix of coefficients. On exit, A is * overwritten by the updated matrix. * * LDA - INTEGER. * On entry, LDA specifies the first dimension of A as declared * in the calling (sub) program. LDA must be at least * max( 1, m ). * Unchanged on exit. * * * Level 2 Blas routine. * * -- Written on 22-October-1986. * Jack Dongarra, Argonne National Lab. * Jeremy Du Croz, Nag Central Office. * Sven Hammarling, Nag Central Office. * Richard Hanson, Sandia National Labs. * * * .. Parameters .. REAL ZERO PARAMETER ( ZERO = 0.0E+0 ) * .. Local Scalars .. REAL TEMP INTEGER I, INFO, IX, J, JY, KX * .. External Subroutines .. EXTERNAL XERBLA * .. Intrinsic Functions .. INTRINSIC MAX * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 IF ( M.LT.0 )THEN INFO = 1 ELSE IF( N.LT.0 )THEN INFO = 2 ELSE IF( INCX.EQ.0 )THEN INFO = 5 ELSE IF( INCY.EQ.0 )THEN INFO = 7 ELSE IF( LDA.LT.MAX( 1, M ) )THEN INFO = 9 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'SGER ', INFO ) RETURN END IF * * Quick return if possible. * IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR.( ALPHA.EQ.ZERO ) ) \$ RETURN * * Start the operations. In this version the elements of A are * accessed sequentially with one pass through A. * IF( INCY.GT.0 )THEN JY = 1 ELSE JY = 1 - ( N - 1 )*INCY END IF IF( INCX.EQ.1 )THEN DO 20, J = 1, N IF( Y( JY ).NE.ZERO )THEN TEMP = ALPHA*Y( JY ) DO 10, I = 1, M A( I, J ) = A( I, J ) + X( I )*TEMP 10 CONTINUE END IF JY = JY + INCY 20 CONTINUE ELSE IF( INCX.GT.0 )THEN KX = 1 ELSE KX = 1 - ( M - 1 )*INCX END IF DO 40, J = 1, N IF( Y( JY ).NE.ZERO )THEN TEMP = ALPHA*Y( JY ) IX = KX DO 30, I = 1, M A( I, J ) = A( I, J ) + X( IX )*TEMP IX = IX + INCX 30 CONTINUE END IF JY = JY + INCY 40 CONTINUE END IF * RETURN * * End of SGER . * END real function snrm2 ( n, sx, incx) integer i, incx, ix, j, n, next real sx(1), cutlo, cuthi, hitest, sum, xmax, zero, one data zero, one /0.0e0, 1.0e0/ c c euclidean norm of the n-vector stored in sx() with storage c increment incx . c if n .le. 0 return with result = 0. c if n .ge. 1 then incx must be .ge. 1 c c c.l.lawson, 1978 jan 08 c modified to correct problem with negative increment, 8/21/90. c modified to correct failure to update ix, 1/25/92. c c four phase method using two built-in constants that are c hopefully applicable to all machines. c cutlo = maximum of sqrt(u/eps) over all known machines. c cuthi = minimum of sqrt(v) over all known machines. c where c eps = smallest no. such that eps + 1. .gt. 1. c u = smallest positive no. (underflow limit) c v = largest no. (overflow limit) c c brief outline of algorithm.. c c phase 1 scans zero components. c move to phase 2 when a component is nonzero and .le. cutlo c move to phase 3 when a component is .gt. cutlo c move to phase 4 when a component is .ge. cuthi/m c where m = n for x() real and m = 2*n for complex. c c values for cutlo and cuthi.. c from the environmental parameters listed in the imsl converter c document the limiting values are as follows.. c cutlo, s.p. u/eps = 2**(-102) for honeywell. close seconds are c univac and dec at 2**(-103) c thus cutlo = 2**(-51) = 4.44089e-16 c cuthi, s.p. v = 2**127 for univac, honeywell, and dec. c thus cuthi = 2**(63.5) = 1.30438e19 c cutlo, d.p. u/eps = 2**(-67) for honeywell and dec. c thus cutlo = 2**(-33.5) = 8.23181d-11 c cuthi, d.p. same as s.p. cuthi = 1.30438d19 c data cutlo, cuthi / 8.232d-11, 1.304d19 / c data cutlo, cuthi / 4.441e-16, 1.304e19 / data cutlo, cuthi / 4.441e-16, 1.304e19 / c if(n .gt. 0) go to 10 snrm2 = zero go to 300 c 10 assign 30 to next sum = zero i = 1 if(incx.lt.0)i = (-n+1)*incx + 1 ix = 1 c begin main loop 20 go to next,(30, 50, 70, 110) 30 if( abs(sx(i)) .gt. cutlo) go to 85 assign 50 to next xmax = zero c c phase 1. sum is zero c 50 if( sx(i) .eq. zero) go to 200 if( abs(sx(i)) .gt. cutlo) go to 85 c c prepare for phase 2. assign 70 to next go to 105 c c prepare for phase 4. c 100 continue ix = j assign 110 to next sum = (sum / sx(i)) / sx(i) 105 xmax = abs(sx(i)) go to 115 c c phase 2. sum is small. c scale to avoid destructive underflow. c 70 if( abs(sx(i)) .gt. cutlo ) go to 75 c c common code for phases 2 and 4. c in phase 4 sum is large. scale to avoid overflow. c 110 if( abs(sx(i)) .le. xmax ) go to 115 sum = one + sum * (xmax / sx(i))**2 xmax = abs(sx(i)) go to 200 c 115 sum = sum + (sx(i)/xmax)**2 go to 200 c c c prepare for phase 3. c 75 sum = (sum * xmax) * xmax c c c for real or d.p. set hitest = cuthi/n c for complex set hitest = cuthi/(2*n) c 85 hitest = cuthi/float( n ) c c phase 3. sum is mid-range. no scaling. c do 95 j = ix, n if(abs(sx(i)) .ge. hitest) go to 100 sum = sum + sx(i)**2 i = i + incx 95 continue snrm2 = sqrt( sum ) go to 300 c 200 continue ix = ix + 1 i = i + incx if( ix .le. n ) go to 20 c c end of main loop. c c compute square root and adjust for scaling. c snrm2 = xmax * sqrt(sum) 300 continue return end subroutine srot (n,sx,incx,sy,incy,c,s) c c applies a plane rotation. c jack dongarra, linpack, 3/11/78. c real sx(1),sy(1),stemp,c,s integer i,incx,incy,ix,iy,n c if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 c c code for unequal increments or equal increments not equal c to 1 c ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n stemp = c*sx(ix) + s*sy(iy) sy(iy) = c*sy(iy) - s*sx(ix) sx(ix) = stemp ix = ix + incx iy = iy + incy 10 continue return c c code for both increments equal to 1 c 20 do 30 i = 1,n stemp = c*sx(i) + s*sy(i) sy(i) = c*sy(i) - s*sx(i) sx(i) = stemp 30 continue return end subroutine sscal(n,sa,sx,incx) c c scales a vector by a constant. c uses unrolled loops for increment equal to 1. c jack dongarra, linpack, 3/11/78. c modified to correct problem with negative increment, 8/21/90. c real sa,sx(1) integer i,incx,ix,m,mp1,n c if(n.le.0)return if(incx.eq.1)go to 20 c c code for increment not equal to 1 c ix = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 do 10 i = 1,n sx(ix) = sa*sx(ix) ix = ix + incx 10 continue return c c code for increment equal to 1 c c c clean-up loop c 20 m = mod(n,5) if( m .eq. 0 ) go to 40 do 30 i = 1,m sx(i) = sa*sx(i) 30 continue if( n .lt. 5 ) return 40 mp1 = m + 1 do 50 i = mp1,n,5 sx(i) = sa*sx(i) sx(i + 1) = sa*sx(i + 1) sx(i + 2) = sa*sx(i + 2) sx(i + 3) = sa*sx(i + 3) sx(i + 4) = sa*sx(i + 4) 50 continue return end subroutine sswap (n,sx,incx,sy,incy) c c interchanges two vectors. c uses unrolled loops for increments equal to 1. c jack dongarra, linpack, 3/11/78. c real sx(1),sy(1),stemp integer i,incx,incy,ix,iy,m,mp1,n c if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 c c code for unequal increments or equal increments not equal c to 1 c ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n stemp = sx(ix) sx(ix) = sy(iy) sy(iy) = stemp ix = ix + incx iy = iy + incy 10 continue return c c code for both increments equal to 1 c c c clean-up loop c 20 m = mod(n,3) if( m .eq. 0 ) go to 40 do 30 i = 1,m stemp = sx(i) sx(i) = sy(i) sy(i) = stemp 30 continue if( n .lt. 3 ) return 40 mp1 = m + 1 do 50 i = mp1,n,3 stemp = sx(i) sx(i) = sy(i) sy(i) = stemp stemp = sx(i + 1) sx(i + 1) = sy(i + 1) sy(i + 1) = stemp stemp = sx(i + 2) sx(i + 2) = sy(i + 2) sy(i + 2) = stemp 50 continue return end SUBROUTINE SSYMM ( SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, \$ BETA, C, LDC ) * .. Scalar Arguments .. CHARACTER*1 SIDE, UPLO INTEGER M, N, LDA, LDB, LDC REAL ALPHA, BETA * .. Array Arguments .. REAL A( LDA, * ), B( LDB, * ), C( LDC, * ) * .. * * Purpose * ======= * * SSYMM performs one of the matrix-matrix operations * * C := alpha*A*B + beta*C, * * or * * C := alpha*B*A + beta*C, * * where alpha and beta are scalars, A is a symmetric matrix and B and * C are m by n matrices. * * Parameters * ========== * * SIDE - CHARACTER*1. * On entry, SIDE specifies whether the symmetric matrix A * appears on the left or right in the operation as follows: * * SIDE = 'L' or 'l' C := alpha*A*B + beta*C, * * SIDE = 'R' or 'r' C := alpha*B*A + beta*C, * * Unchanged on exit. * * UPLO - CHARACTER*1. * On entry, UPLO specifies whether the upper or lower * triangular part of the symmetric matrix A is to be * referenced as follows: * * UPLO = 'U' or 'u' Only the upper triangular part of the * symmetric matrix is to be referenced. * * UPLO = 'L' or 'l' Only the lower triangular part of the * symmetric matrix is to be referenced. * * Unchanged on exit. * * M - INTEGER. * On entry, M specifies the number of rows of the matrix C. * M must be at least zero. * Unchanged on exit. * * N - INTEGER. * On entry, N specifies the number of columns of the matrix C. * N must be at least zero. * Unchanged on exit. * * ALPHA - REAL . * On entry, ALPHA specifies the scalar alpha. * Unchanged on exit. * * A - REAL array of DIMENSION ( LDA, ka ), where ka is * m when SIDE = 'L' or 'l' and is n otherwise. * Before entry with SIDE = 'L' or 'l', the m by m part of * the array A must contain the symmetric matrix, such that * when UPLO = 'U' or 'u', the leading m by m upper triangular * part of the array A must contain the upper triangular part * of the symmetric matrix and the strictly lower triangular * part of A is not referenced, and when UPLO = 'L' or 'l', * the leading m by m lower triangular part of the array A * must contain the lower triangular part of the symmetric * matrix and the strictly upper triangular part of A is not * referenced. * Before entry with SIDE = 'R' or 'r', the n by n part of * the array A must contain the symmetric matrix, such that * when UPLO = 'U' or 'u', the leading n by n upper triangular * part of the array A must contain the upper triangular part * of the symmetric matrix and the strictly lower triangular * part of A is not referenced, and when UPLO = 'L' or 'l', * the leading n by n lower triangular part of the array A * must contain the lower triangular part of the symmetric * matrix and the strictly upper triangular part of A is not * referenced. * Unchanged on exit. * * LDA - INTEGER. * On entry, LDA specifies the first dimension of A as declared * in the calling (sub) program. When SIDE = 'L' or 'l' then * LDA must be at least max( 1, m ), otherwise LDA must be at * least max( 1, n ). * Unchanged on exit. * * B - REAL array of DIMENSION ( LDB, n ). * Before entry, the leading m by n part of the array B must * contain the matrix B. * Unchanged on exit. * * LDB - INTEGER. * On entry, LDB specifies the first dimension of B as declared * in the calling (sub) program. LDB must be at least * max( 1, m ). * Unchanged on exit. * * BETA - REAL . * On entry, BETA specifies the scalar beta. When BETA is * supplied as zero then C need not be set on input. * Unchanged on exit. * * C - REAL array of DIMENSION ( LDC, n ). * Before entry, the leading m by n part of the array C must * contain the matrix C, except when beta is zero, in which * case C need not be set on entry. * On exit, the array C is overwritten by the m by n updated * matrix. * * LDC - INTEGER. * On entry, LDC specifies the first dimension of C as declared * in the calling (sub) program. LDC must be at least * max( 1, m ). * Unchanged on exit. * * * Level 3 Blas routine. * * -- Written on 8-February-1989. * Jack Dongarra, Argonne National Laboratory. * Iain Duff, AERE Harwell. * Jeremy Du Croz, Numerical Algorithms Group Ltd. * Sven Hammarling, Numerical Algorithms Group Ltd. * * * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. External Subroutines .. EXTERNAL XERBLA * .. Intrinsic Functions .. INTRINSIC MAX * .. Local Scalars .. LOGICAL UPPER INTEGER I, INFO, J, K, NROWA REAL TEMP1, TEMP2 * .. Parameters .. REAL ONE , ZERO PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) * .. * .. Executable Statements .. * * Set NROWA as the number of rows of A. * IF( LSAME( SIDE, 'L' ) )THEN NROWA = M ELSE NROWA = N END IF UPPER = LSAME( UPLO, 'U' ) * * Test the input parameters. * INFO = 0 IF( ( .NOT.LSAME( SIDE, 'L' ) ).AND. \$ ( .NOT.LSAME( SIDE, 'R' ) ) )THEN INFO = 1 ELSE IF( ( .NOT.UPPER ).AND. \$ ( .NOT.LSAME( UPLO, 'L' ) ) )THEN INFO = 2 ELSE IF( M .LT.0 )THEN INFO = 3 ELSE IF( N .LT.0 )THEN INFO = 4 ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN INFO = 7 ELSE IF( LDB.LT.MAX( 1, M ) )THEN INFO = 9 ELSE IF( LDC.LT.MAX( 1, M ) )THEN INFO = 12 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'SSYMM ', INFO ) RETURN END IF * * Quick return if possible. * IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR. \$ ( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) ) \$ RETURN * * And when alpha.eq.zero. * IF( ALPHA.EQ.ZERO )THEN IF( BETA.EQ.ZERO )THEN DO 20, J = 1, N DO 10, I = 1, M C( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ELSE DO 40, J = 1, N DO 30, I = 1, M C( I, J ) = BETA*C( I, J ) 30 CONTINUE 40 CONTINUE END IF RETURN END IF * * Start the operations. * IF( LSAME( SIDE, 'L' ) )THEN * * Form C := alpha*A*B + beta*C. * IF( UPPER )THEN DO 70, J = 1, N DO 60, I = 1, M TEMP1 = ALPHA*B( I, J ) TEMP2 = ZERO DO 50, K = 1, I - 1 C( K, J ) = C( K, J ) + TEMP1 *A( K, I ) TEMP2 = TEMP2 + B( K, J )*A( K, I ) 50 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = TEMP1*A( I, I ) + ALPHA*TEMP2 ELSE C( I, J ) = BETA *C( I, J ) + \$ TEMP1*A( I, I ) + ALPHA*TEMP2 END IF 60 CONTINUE 70 CONTINUE ELSE DO 100, J = 1, N DO 90, I = M, 1, -1 TEMP1 = ALPHA*B( I, J ) TEMP2 = ZERO DO 80, K = I + 1, M C( K, J ) = C( K, J ) + TEMP1 *A( K, I ) TEMP2 = TEMP2 + B( K, J )*A( K, I ) 80 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = TEMP1*A( I, I ) + ALPHA*TEMP2 ELSE C( I, J ) = BETA *C( I, J ) + \$ TEMP1*A( I, I ) + ALPHA*TEMP2 END IF 90 CONTINUE 100 CONTINUE END IF ELSE * * Form C := alpha*B*A + beta*C. * DO 170, J = 1, N TEMP1 = ALPHA*A( J, J ) IF( BETA.EQ.ZERO )THEN DO 110, I = 1, M C( I, J ) = TEMP1*B( I, J ) 110 CONTINUE ELSE DO 120, I = 1, M C( I, J ) = BETA*C( I, J ) + TEMP1*B( I, J ) 120 CONTINUE END IF DO 140, K = 1, J - 1 IF( UPPER )THEN TEMP1 = ALPHA*A( K, J ) ELSE TEMP1 = ALPHA*A( J, K ) END IF DO 130, I = 1, M C( I, J ) = C( I, J ) + TEMP1*B( I, K ) 130 CONTINUE 140 CONTINUE DO 160, K = J + 1, N IF( UPPER )THEN TEMP1 = ALPHA*A( J, K ) ELSE TEMP1 = ALPHA*A( K, J ) END IF DO 150, I = 1, M C( I, J ) = C( I, J ) + TEMP1*B( I, K ) 150 CONTINUE 160 CONTINUE 170 CONTINUE END IF * RETURN * * End of SSYMM . * END * ************************************************************************ * SUBROUTINE SSYR2 ( UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA ) * .. Scalar Arguments .. REAL ALPHA INTEGER INCX, INCY, LDA, N CHARACTER*1 UPLO * .. Array Arguments .. REAL A( LDA, * ), X( * ), Y( * ) * .. * * Purpose * ======= * * SSYR2 performs the symmetric rank 2 operation * * A := alpha*x*y' + alpha*y*x' + A, * * where alpha is a scalar, x and y are n element vectors and A is an n * by n symmetric matrix. * * Parameters * ========== * * UPLO - CHARACTER*1. * On entry, UPLO specifies whether the upper or lower * triangular part of the array A is to be referenced as * follows: * * UPLO = 'U' or 'u' Only the upper triangular part of A * is to be referenced. * * UPLO = 'L' or 'l' Only the lower triangular part of A * is to be referenced. * * Unchanged on exit. * * N - INTEGER. * On entry, N specifies the order of the matrix A. * N must be at least zero. * Unchanged on exit. * * ALPHA - REAL . * On entry, ALPHA specifies the scalar alpha. * Unchanged on exit. * * X - REAL array of dimension at least * ( 1 + ( n - 1 )*abs( INCX ) ). * Before entry, the incremented array X must contain the n * element vector x. * Unchanged on exit. * * INCX - INTEGER. * On entry, INCX specifies the increment for the elements of * X. INCX must not be zero. * Unchanged on exit. * * Y - REAL array of dimension at least * ( 1 + ( n - 1 )*abs( INCY ) ). * Before entry, the incremented array Y must contain the n * element vector y. * Unchanged on exit. * * INCY - INTEGER. * On entry, INCY specifies the increment for the elements of * Y. INCY must not be zero. * Unchanged on exit. * * A - REAL array of DIMENSION ( LDA, n ). * Before entry with UPLO = 'U' or 'u', the leading n by n * upper triangular part of the array A must contain the upper * triangular part of the symmetric matrix and the strictly * lower triangular part of A is not referenced. On exit, the * upper triangular part of the array A is overwritten by the * upper triangular part of the updated matrix. * Before entry with UPLO = 'L' or 'l', the leading n by n * lower triangular part of the array A must contain the lower * triangular part of the symmetric matrix and the strictly * upper triangular part of A is not referenced. On exit, the * lower triangular part of the array A is overwritten by the * lower triangular part of the updated matrix. * * LDA - INTEGER. * On entry, LDA specifies the first dimension of A as declared * in the calling (sub) program. LDA must be at least * max( 1, n ). * Unchanged on exit. * * * Level 2 Blas routine. * * -- Written on 22-October-1986. * Jack Dongarra, Argonne National Lab. * Jeremy Du Croz, Nag Central Office. * Sven Hammarling, Nag Central Office. * Richard Hanson, Sandia National Labs. * * * .. Parameters .. REAL ZERO PARAMETER ( ZERO = 0.0E+0 ) * .. Local Scalars .. REAL TEMP1, TEMP2 INTEGER I, INFO, IX, IY, J, JX, JY, KX, KY * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. External Subroutines .. EXTERNAL XERBLA * .. Intrinsic Functions .. INTRINSIC MAX * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 IF ( .NOT.LSAME( UPLO, 'U' ).AND. \$ .NOT.LSAME( UPLO, 'L' ) )THEN INFO = 1 ELSE IF( N.LT.0 )THEN INFO = 2 ELSE IF( INCX.EQ.0 )THEN INFO = 5 ELSE IF( INCY.EQ.0 )THEN INFO = 7 ELSE IF( LDA.LT.MAX( 1, N ) )THEN INFO = 9 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'SSYR2 ', INFO ) RETURN END IF * * Quick return if possible. * IF( ( N.EQ.0 ).OR.( ALPHA.EQ.ZERO ) ) \$ RETURN * * Set up the start points in X and Y if the increments are not both * unity. * IF( ( INCX.NE.1 ).OR.( INCY.NE.1 ) )THEN IF( INCX.GT.0 )THEN KX = 1 ELSE KX = 1 - ( N - 1 )*INCX END IF IF( INCY.GT.0 )THEN KY = 1 ELSE KY = 1 - ( N - 1 )*INCY END IF JX = KX JY = KY END IF * * Start the operations. In this version the elements of A are * accessed sequentially with one pass through the triangular part * of A. * IF( LSAME( UPLO, 'U' ) )THEN * * Form A when A is stored in the upper triangle. * IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN DO 20, J = 1, N IF( ( X( J ).NE.ZERO ).OR.( Y( J ).NE.ZERO ) )THEN TEMP1 = ALPHA*Y( J ) TEMP2 = ALPHA*X( J ) DO 10, I = 1, J A( I, J ) = A( I, J ) + X( I )*TEMP1 + Y( I )*TEMP2 10 CONTINUE END IF 20 CONTINUE ELSE DO 40, J = 1, N IF( ( X( JX ).NE.ZERO ).OR.( Y( JY ).NE.ZERO ) )THEN TEMP1 = ALPHA*Y( JY ) TEMP2 = ALPHA*X( JX ) IX = KX IY = KY DO 30, I = 1, J A( I, J ) = A( I, J ) + X( IX )*TEMP1 \$ + Y( IY )*TEMP2 IX = IX + INCX IY = IY + INCY 30 CONTINUE END IF JX = JX + INCX JY = JY + INCY 40 CONTINUE END IF ELSE * * Form A when A is stored in the lower triangle. * IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN DO 60, J = 1, N IF( ( X( J ).NE.ZERO ).OR.( Y( J ).NE.ZERO ) )THEN TEMP1 = ALPHA*Y( J ) TEMP2 = ALPHA*X( J ) DO 50, I = J, N A( I, J ) = A( I, J ) + X( I )*TEMP1 + Y( I )*TEMP2 50 CONTINUE END IF 60 CONTINUE ELSE DO 80, J = 1, N IF( ( X( JX ).NE.ZERO ).OR.( Y( JY ).NE.ZERO ) )THEN TEMP1 = ALPHA*Y( JY ) TEMP2 = ALPHA*X( JX ) IX = JX IY = JY DO 70, I = J, N A( I, J ) = A( I, J ) + X( IX )*TEMP1 \$ + Y( IY )*TEMP2 IX = IX + INCX IY = IY + INCY 70 CONTINUE END IF JX = JX + INCX JY = JY + INCY 80 CONTINUE END IF END IF * RETURN * * End of SSYR2 . * END SUBROUTINE SSYR2K( UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, \$ BETA, C, LDC ) * .. Scalar Arguments .. CHARACTER*1 UPLO, TRANS INTEGER N, K, LDA, LDB, LDC REAL ALPHA, BETA * .. Array Arguments .. REAL A( LDA, * ), B( LDB, * ), C( LDC, * ) * .. * * Purpose * ======= * * SSYR2K performs one of the symmetric rank 2k operations * * C := alpha*A*B' + alpha*B*A' + beta*C, * * or * * C := alpha*A'*B + alpha*B'*A + beta*C, * * where alpha and beta are scalars, C is an n by n symmetric matrix * and A and B are n by k matrices in the first case and k by n * matrices in the second case. * * Parameters * ========== * * UPLO - CHARACTER*1. * On entry, UPLO specifies whether the upper or lower * triangular part of the array C is to be referenced as * follows: * * UPLO = 'U' or 'u' Only the upper triangular part of C * is to be referenced. * * UPLO = 'L' or 'l' Only the lower triangular part of C * is to be referenced. * * Unchanged on exit. * * TRANS - CHARACTER*1. * On entry, TRANS specifies the operation to be performed as * follows: * * TRANS = 'N' or 'n' C := alpha*A*B' + alpha*B*A' + * beta*C. * * TRANS = 'T' or 't' C := alpha*A'*B + alpha*B'*A + * beta*C. * * TRANS = 'C' or 'c' C := alpha*A'*B + alpha*B'*A + * beta*C. * * Unchanged on exit. * * N - INTEGER. * On entry, N specifies the order of the matrix C. N must be * at least zero. * Unchanged on exit. * * K - INTEGER. * On entry with TRANS = 'N' or 'n', K specifies the number * of columns of the matrices A and B, and on entry with * TRANS = 'T' or 't' or 'C' or 'c', K specifies the number * of rows of the matrices A and B. K must be at least zero. * Unchanged on exit. * * ALPHA - REAL . * On entry, ALPHA specifies the scalar alpha. * Unchanged on exit. * * A - REAL array of DIMENSION ( LDA, ka ), where ka is * k when TRANS = 'N' or 'n', and is n otherwise. * Before entry with TRANS = 'N' or 'n', the leading n by k * part of the array A must contain the matrix A, otherwise * the leading k by n part of the array A must contain the * matrix A. * Unchanged on exit. * * LDA - INTEGER. * On entry, LDA specifies the first dimension of A as declared * in the calling (sub) program. When TRANS = 'N' or 'n' * then LDA must be at least max( 1, n ), otherwise LDA must * be at least max( 1, k ). * Unchanged on exit. * * B - REAL array of DIMENSION ( LDB, kb ), where kb is * k when TRANS = 'N' or 'n', and is n otherwise. * Before entry with TRANS = 'N' or 'n', the leading n by k * part of the array B must contain the matrix B, otherwise * the leading k by n part of the array B must contain the * matrix B. * Unchanged on exit. * * LDB - INTEGER. * On entry, LDB specifies the first dimension of B as declared * in the calling (sub) program. When TRANS = 'N' or 'n' * then LDB must be at least max( 1, n ), otherwise LDB must * be at least max( 1, k ). * Unchanged on exit. * * BETA - REAL . * On entry, BETA specifies the scalar beta. * Unchanged on exit. * * C - REAL array of DIMENSION ( LDC, n ). * Before entry with UPLO = 'U' or 'u', the leading n by n * upper triangular part of the array C must contain the upper * triangular part of the symmetric matrix and the strictly * lower triangular part of C is not referenced. On exit, the * upper triangular part of the array C is overwritten by the * upper triangular part of the updated matrix. * Before entry with UPLO = 'L' or 'l', the leading n by n * lower triangular part of the array C must contain the lower * triangular part of the symmetric matrix and the strictly * upper triangular part of C is not referenced. On exit, the * lower triangular part of the array C is overwritten by the * lower triangular part of the updated matrix. * * LDC - INTEGER. * On entry, LDC specifies the first dimension of C as declared * in the calling (sub) program. LDC must be at least * max( 1, n ). * Unchanged on exit. * * * Level 3 Blas routine. * * * -- Written on 8-February-1989. * Jack Dongarra, Argonne National Laboratory. * Iain Duff, AERE Harwell. * Jeremy Du Croz, Numerical Algorithms Group Ltd. * Sven Hammarling, Numerical Algorithms Group Ltd. * * * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. External Subroutines .. EXTERNAL XERBLA * .. Intrinsic Functions .. INTRINSIC MAX * .. Local Scalars .. LOGICAL UPPER INTEGER I, INFO, J, L, NROWA REAL TEMP1, TEMP2 * .. Parameters .. REAL ONE , ZERO PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) * .. * .. Executable Statements .. * * Test the input parameters. * IF( LSAME( TRANS, 'N' ) )THEN NROWA = N ELSE NROWA = K END IF UPPER = LSAME( UPLO, 'U' ) * INFO = 0 IF( ( .NOT.UPPER ).AND. \$ ( .NOT.LSAME( UPLO , 'L' ) ) )THEN INFO = 1 ELSE IF( ( .NOT.LSAME( TRANS, 'N' ) ).AND. \$ ( .NOT.LSAME( TRANS, 'T' ) ).AND. \$ ( .NOT.LSAME( TRANS, 'C' ) ) )THEN INFO = 2 ELSE IF( N .LT.0 )THEN INFO = 3 ELSE IF( K .LT.0 )THEN INFO = 4 ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN INFO = 7 ELSE IF( LDB.LT.MAX( 1, NROWA ) )THEN INFO = 9 ELSE IF( LDC.LT.MAX( 1, N ) )THEN INFO = 12 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'SSYR2K', INFO ) RETURN END IF * * Quick return if possible. * IF( ( N.EQ.0 ).OR. \$ ( ( ( ALPHA.EQ.ZERO ).OR.( K.EQ.0 ) ).AND.( BETA.EQ.ONE ) ) ) \$ RETURN * * And when alpha.eq.zero. * IF( ALPHA.EQ.ZERO )THEN IF( UPPER )THEN IF( BETA.EQ.ZERO )THEN DO 20, J = 1, N DO 10, I = 1, J C( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ELSE DO 40, J = 1, N DO 30, I = 1, J C( I, J ) = BETA*C( I, J ) 30 CONTINUE 40 CONTINUE END IF ELSE IF( BETA.EQ.ZERO )THEN DO 60, J = 1, N DO 50, I = J, N C( I, J ) = ZERO 50 CONTINUE 60 CONTINUE ELSE DO 80, J = 1, N DO 70, I = J, N C( I, J ) = BETA*C( I, J ) 70 CONTINUE 80 CONTINUE END IF END IF RETURN END IF * * Start the operations. * IF( LSAME( TRANS, 'N' ) )THEN * * Form C := alpha*A*B' + alpha*B*A' + C. * IF( UPPER )THEN DO 130, J = 1, N IF( BETA.EQ.ZERO )THEN DO 90, I = 1, J C( I, J ) = ZERO 90 CONTINUE ELSE IF( BETA.NE.ONE )THEN DO 100, I = 1, J C( I, J ) = BETA*C( I, J ) 100 CONTINUE END IF DO 120, L = 1, K IF( ( A( J, L ).NE.ZERO ).OR. \$ ( B( J, L ).NE.ZERO ) )THEN TEMP1 = ALPHA*B( J, L ) TEMP2 = ALPHA*A( J, L ) DO 110, I = 1, J C( I, J ) = C( I, J ) + \$ A( I, L )*TEMP1 + B( I, L )*TEMP2 110 CONTINUE END IF 120 CONTINUE 130 CONTINUE ELSE DO 180, J = 1, N IF( BETA.EQ.ZERO )THEN DO 140, I = J, N C( I, J ) = ZERO 140 CONTINUE ELSE IF( BETA.NE.ONE )THEN DO 150, I = J, N C( I, J ) = BETA*C( I, J ) 150 CONTINUE END IF DO 170, L = 1, K IF( ( A( J, L ).NE.ZERO ).OR. \$ ( B( J, L ).NE.ZERO ) )THEN TEMP1 = ALPHA*B( J, L ) TEMP2 = ALPHA*A( J, L ) DO 160, I = J, N C( I, J ) = C( I, J ) + \$ A( I, L )*TEMP1 + B( I, L )*TEMP2 160 CONTINUE END IF 170 CONTINUE 180 CONTINUE END IF ELSE * * Form C := alpha*A'*B + alpha*B'*A + C. * IF( UPPER )THEN DO 210, J = 1, N DO 200, I = 1, J TEMP1 = ZERO TEMP2 = ZERO DO 190, L = 1, K TEMP1 = TEMP1 + A( L, I )*B( L, J ) TEMP2 = TEMP2 + B( L, I )*A( L, J ) 190 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = ALPHA*TEMP1 + ALPHA*TEMP2 ELSE C( I, J ) = BETA *C( I, J ) + \$ ALPHA*TEMP1 + ALPHA*TEMP2 END IF 200 CONTINUE 210 CONTINUE ELSE DO 240, J = 1, N DO 230, I = J, N TEMP1 = ZERO TEMP2 = ZERO DO 220, L = 1, K TEMP1 = TEMP1 + A( L, I )*B( L, J ) TEMP2 = TEMP2 + B( L, I )*A( L, J ) 220 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = ALPHA*TEMP1 + ALPHA*TEMP2 ELSE C( I, J ) = BETA *C( I, J ) + \$ ALPHA*TEMP1 + ALPHA*TEMP2 END IF 230 CONTINUE 240 CONTINUE END IF END IF * RETURN * * End of SSYR2K. * END SUBROUTINE STRMM ( SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, \$ B, LDB ) * .. Scalar Arguments .. CHARACTER*1 SIDE, UPLO, TRANSA, DIAG INTEGER M, N, LDA, LDB REAL ALPHA * .. Array Arguments .. REAL A( LDA, * ), B( LDB, * ) * .. * * Purpose * ======= * * STRMM performs one of the matrix-matrix operations * * B := alpha*op( A )*B, or B := alpha*B*op( A ), * * where alpha is a scalar, B is an m by n matrix, A is a unit, or * non-unit, upper or lower triangular matrix and op( A ) is one of * * op( A ) = A or op( A ) = A'. * * Parameters * ========== * * SIDE - CHARACTER*1. * On entry, SIDE specifies whether op( A ) multiplies B from * the left or right as follows: * * SIDE = 'L' or 'l' B := alpha*op( A )*B. * * SIDE = 'R' or 'r' B := alpha*B*op( A ). * * Unchanged on exit. * * UPLO - CHARACTER*1. * On entry, UPLO specifies whether the matrix A is an upper or * lower triangular matrix as follows: * * UPLO = 'U' or 'u' A is an upper triangular matrix. * * UPLO = 'L' or 'l' A is a lower triangular matrix. * * Unchanged on exit. * * TRANSA - CHARACTER*1. * On entry, TRANSA specifies the form of op( A ) to be used in * the matrix multiplication as follows: * * TRANSA = 'N' or 'n' op( A ) = A. * * TRANSA = 'T' or 't' op( A ) = A'. * * TRANSA = 'C' or 'c' op( A ) = A'. * * Unchanged on exit. * * DIAG - CHARACTER*1. * On entry, DIAG specifies whether or not A is unit triangular * as follows: * * DIAG = 'U' or 'u' A is assumed to be unit triangular. * * DIAG = 'N' or 'n' A is not assumed to be unit * triangular. * * Unchanged on exit. * * M - INTEGER. * On entry, M specifies the number of rows of B. M must be at * least zero. * Unchanged on exit. * * N - INTEGER. * On entry, N specifies the number of columns of B. N must be * at least zero. * Unchanged on exit. * * ALPHA - REAL . * On entry, ALPHA specifies the scalar alpha. When alpha is * zero then A is not referenced and B need not be set before * entry. * Unchanged on exit. * * A - REAL array of DIMENSION ( LDA, k ), where k is m * when SIDE = 'L' or 'l' and is n when SIDE = 'R' or 'r'. * Before entry with UPLO = 'U' or 'u', the leading k by k * upper triangular part of the array A must contain the upper * triangular matrix and the strictly lower triangular part of * A is not referenced. * Before entry with UPLO = 'L' or 'l', the leading k by k * lower triangular part of the array A must contain the lower * triangular matrix and the strictly upper triangular part of * A is not referenced. * Note that when DIAG = 'U' or 'u', the diagonal elements of * A are not referenced either, but are assumed to be unity. * Unchanged on exit. * * LDA - INTEGER. * On entry, LDA specifies the first dimension of A as declared * in the calling (sub) program. When SIDE = 'L' or 'l' then * LDA must be at least max( 1, m ), when SIDE = 'R' or 'r' * then LDA must be at least max( 1, n ). * Unchanged on exit. * * B - REAL array of DIMENSION ( LDB, n ). * Before entry, the leading m by n part of the array B must * contain the matrix B, and on exit is overwritten by the * transformed matrix. * * LDB - INTEGER. * On entry, LDB specifies the first dimension of B as declared * in the calling (sub) program. LDB must be at least * max( 1, m ). * Unchanged on exit. * * * Level 3 Blas routine. * * -- Written on 8-February-1989. * Jack Dongarra, Argonne National Laboratory. * Iain Duff, AERE Harwell. * Jeremy Du Croz, Numerical Algorithms Group Ltd. * Sven Hammarling, Numerical Algorithms Group Ltd. * * * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. External Subroutines .. EXTERNAL XERBLA * .. Intrinsic Functions .. INTRINSIC MAX * .. Local Scalars .. LOGICAL LSIDE, NOUNIT, UPPER INTEGER I, INFO, J, K, NROWA REAL TEMP * .. Parameters .. REAL ONE , ZERO PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) * .. * .. Executable Statements .. * * Test the input parameters. * LSIDE = LSAME( SIDE , 'L' ) IF( LSIDE )THEN NROWA = M ELSE NROWA = N END IF NOUNIT = LSAME( DIAG , 'N' ) UPPER = LSAME( UPLO , 'U' ) * INFO = 0 IF( ( .NOT.LSIDE ).AND. \$ ( .NOT.LSAME( SIDE , 'R' ) ) )THEN INFO = 1 ELSE IF( ( .NOT.UPPER ).AND. \$ ( .NOT.LSAME( UPLO , 'L' ) ) )THEN INFO = 2 ELSE IF( ( .NOT.LSAME( TRANSA, 'N' ) ).AND. \$ ( .NOT.LSAME( TRANSA, 'T' ) ).AND. \$ ( .NOT.LSAME( TRANSA, 'C' ) ) )THEN INFO = 3 ELSE IF( ( .NOT.LSAME( DIAG , 'U' ) ).AND. \$ ( .NOT.LSAME( DIAG , 'N' ) ) )THEN INFO = 4 ELSE IF( M .LT.0 )THEN INFO = 5 ELSE IF( N .LT.0 )THEN INFO = 6 ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN INFO = 9 ELSE IF( LDB.LT.MAX( 1, M ) )THEN INFO = 11 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'STRMM ', INFO ) RETURN END IF * * Quick return if possible. * IF( N.EQ.0 ) \$ RETURN * * And when alpha.eq.zero. * IF( ALPHA.EQ.ZERO )THEN DO 20, J = 1, N DO 10, I = 1, M B( I, J ) = ZERO 10 CONTINUE 20 CONTINUE RETURN END IF * * Start the operations. * IF( LSIDE )THEN IF( LSAME( TRANSA, 'N' ) )THEN * * Form B := alpha*A*B. * IF( UPPER )THEN DO 50, J = 1, N DO 40, K = 1, M IF( B( K, J ).NE.ZERO )THEN TEMP = ALPHA*B( K, J ) DO 30, I = 1, K - 1 B( I, J ) = B( I, J ) + TEMP*A( I, K ) 30 CONTINUE IF( NOUNIT ) \$ TEMP = TEMP*A( K, K ) B( K, J ) = TEMP END IF 40 CONTINUE 50 CONTINUE ELSE DO 80, J = 1, N DO 70 K = M, 1, -1 IF( B( K, J ).NE.ZERO )THEN TEMP = ALPHA*B( K, J ) B( K, J ) = TEMP IF( NOUNIT ) \$ B( K, J ) = B( K, J )*A( K, K ) DO 60, I = K + 1, M B( I, J ) = B( I, J ) + TEMP*A( I, K ) 60 CONTINUE END IF 70 CONTINUE 80 CONTINUE END IF ELSE * * Form B := alpha*B*A'. * IF( UPPER )THEN DO 110, J = 1, N DO 100, I = M, 1, -1 TEMP = B( I, J ) IF( NOUNIT ) \$ TEMP = TEMP*A( I, I ) DO 90, K = 1, I - 1 TEMP = TEMP + A( K, I )*B( K, J ) 90 CONTINUE B( I, J ) = ALPHA*TEMP 100 CONTINUE 110 CONTINUE ELSE DO 140, J = 1, N DO 130, I = 1, M TEMP = B( I, J ) IF( NOUNIT ) \$ TEMP = TEMP*A( I, I ) DO 120, K = I + 1, M TEMP = TEMP + A( K, I )*B( K, J ) 120 CONTINUE B( I, J ) = ALPHA*TEMP 130 CONTINUE 140 CONTINUE END IF END IF ELSE IF( LSAME( TRANSA, 'N' ) )THEN * * Form B := alpha*B*A. * IF( UPPER )THEN DO 180, J = N, 1, -1 TEMP = ALPHA IF( NOUNIT ) \$ TEMP = TEMP*A( J, J ) DO 150, I = 1, M B( I, J ) = TEMP*B( I, J ) 150 CONTINUE DO 170, K = 1, J - 1 IF( A( K, J ).NE.ZERO )THEN TEMP = ALPHA*A( K, J ) DO 160, I = 1, M B( I, J ) = B( I, J ) + TEMP*B( I, K ) 160 CONTINUE END IF 170 CONTINUE 180 CONTINUE ELSE DO 220, J = 1, N TEMP = ALPHA IF( NOUNIT ) \$ TEMP = TEMP*A( J, J ) DO 190, I = 1, M B( I, J ) = TEMP*B( I, J ) 190 CONTINUE DO 210, K = J + 1, N IF( A( K, J ).NE.ZERO )THEN TEMP = ALPHA*A( K, J ) DO 200, I = 1, M B( I, J ) = B( I, J ) + TEMP*B( I, K ) 200 CONTINUE END IF 210 CONTINUE 220 CONTINUE END IF ELSE * * Form B := alpha*B*A'. * IF( UPPER )THEN DO 260, K = 1, N DO 240, J = 1, K - 1 IF( A( J, K ).NE.ZERO )THEN TEMP = ALPHA*A( J, K ) DO 230, I = 1, M B( I, J ) = B( I, J ) + TEMP*B( I, K ) 230 CONTINUE END IF 240 CONTINUE TEMP = ALPHA IF( NOUNIT ) \$ TEMP = TEMP*A( K, K ) IF( TEMP.NE.ONE )THEN DO 250, I = 1, M B( I, K ) = TEMP*B( I, K ) 250 CONTINUE END IF 260 CONTINUE ELSE DO 300, K = N, 1, -1 DO 280, J = K + 1, N IF( A( J, K ).NE.ZERO )THEN TEMP = ALPHA*A( J, K ) DO 270, I = 1, M B( I, J ) = B( I, J ) + TEMP*B( I, K ) 270 CONTINUE END IF 280 CONTINUE TEMP = ALPHA IF( NOUNIT ) \$ TEMP = TEMP*A( K, K ) IF( TEMP.NE.ONE )THEN DO 290, I = 1, M B( I, K ) = TEMP*B( I, K ) 290 CONTINUE END IF 300 CONTINUE END IF END IF END IF * RETURN * * End of STRMM . * END * ************************************************************************ * SUBROUTINE STRMV ( UPLO, TRANS, DIAG, N, A, LDA, X, INCX ) * .. Scalar Arguments .. INTEGER INCX, LDA, N CHARACTER*1 DIAG, TRANS, UPLO * .. Array Arguments .. REAL A( LDA, * ), X( * ) * .. * * Purpose * ======= * * STRMV performs one of the matrix-vector operations * * x := A*x, or x := A'*x, * * where x is an n element vector and A is an n by n unit, or non-unit, * upper or lower triangular matrix. * * Parameters * ========== * * UPLO - CHARACTER*1. * On entry, UPLO specifies whether the matrix is an upper or * lower triangular matrix as follows: * * UPLO = 'U' or 'u' A is an upper triangular matrix. * * UPLO = 'L' or 'l' A is a lower triangular matrix. * * Unchanged on exit. * * TRANS - CHARACTER*1. * On entry, TRANS specifies the operation to be performed as * follows: * * TRANS = 'N' or 'n' x := A*x. * * TRANS = 'T' or 't' x := A'*x. * * TRANS = 'C' or 'c' x := A'*x. * * Unchanged on exit. * * DIAG - CHARACTER*1. * On entry, DIAG specifies whether or not A is unit * triangular as follows: * * DIAG = 'U' or 'u' A is assumed to be unit triangular. * * DIAG = 'N' or 'n' A is not assumed to be unit * triangular. * * Unchanged on exit. * * N - INTEGER. * On entry, N specifies the order of the matrix A. * N must be at least zero. * Unchanged on exit. * * A - REAL array of DIMENSION ( LDA, n ). * Before entry with UPLO = 'U' or 'u', the leading n by n * upper triangular part of the array A must contain the upper * triangular matrix and the strictly lower triangular part of * A is not referenced. * Before entry with UPLO = 'L' or 'l', the leading n by n * lower triangular part of the array A must contain the lower * triangular matrix and the strictly upper triangular part of * A is not referenced. * Note that when DIAG = 'U' or 'u', the diagonal elements of * A are not referenced either, but are assumed to be unity. * Unchanged on exit. * * LDA - INTEGER. * On entry, LDA specifies the first dimension of A as declared * in the calling (sub) program. LDA must be at least * max( 1, n ). * Unchanged on exit. * * X - REAL array of dimension at least * ( 1 + ( n - 1 )*abs( INCX ) ). * Before entry, the incremented array X must contain the n * element vector x. On exit, X is overwritten with the * tranformed vector x. * * INCX - INTEGER. * On entry, INCX specifies the increment for the elements of * X. INCX must not be zero. * Unchanged on exit. * * * Level 2 Blas routine. * * -- Written on 22-October-1986. * Jack Dongarra, Argonne National Lab. * Jeremy Du Croz, Nag Central Office. * Sven Hammarling, Nag Central Office. * Richard Hanson, Sandia National Labs. * * * .. Parameters .. REAL ZERO PARAMETER ( ZERO = 0.0E+0 ) * .. Local Scalars .. REAL TEMP INTEGER I, INFO, IX, J, JX, KX LOGICAL NOUNIT * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. External Subroutines .. EXTERNAL XERBLA * .. Intrinsic Functions .. INTRINSIC MAX * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 IF ( .NOT.LSAME( UPLO , 'U' ).AND. \$ .NOT.LSAME( UPLO , 'L' ) )THEN INFO = 1 ELSE IF( .NOT.LSAME( TRANS, 'N' ).AND. \$ .NOT.LSAME( TRANS, 'T' ).AND. \$ .NOT.LSAME( TRANS, 'C' ) )THEN INFO = 2 ELSE IF( .NOT.LSAME( DIAG , 'U' ).AND. \$ .NOT.LSAME( DIAG , 'N' ) )THEN INFO = 3 ELSE IF( N.LT.0 )THEN INFO = 4 ELSE IF( LDA.LT.MAX( 1, N ) )THEN INFO = 6 ELSE IF( INCX.EQ.0 )THEN INFO = 8 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'STRMV ', INFO ) RETURN END IF * * Quick return if possible. * IF( N.EQ.0 ) \$ RETURN * NOUNIT = LSAME( DIAG, 'N' ) * * Set up the start point in X if the increment is not unity. This * will be ( N - 1 )*INCX too small for descending loops. * IF( INCX.LE.0 )THEN KX = 1 - ( N - 1 )*INCX ELSE IF( INCX.NE.1 )THEN KX = 1 END IF * * Start the operations. In this version the elements of A are * accessed sequentially with one pass through A. * IF( LSAME( TRANS, 'N' ) )THEN * * Form x := A*x. * IF( LSAME( UPLO, 'U' ) )THEN IF( INCX.EQ.1 )THEN DO 20, J = 1, N IF( X( J ).NE.ZERO )THEN TEMP = X( J ) DO 10, I = 1, J - 1 X( I ) = X( I ) + TEMP*A( I, J ) 10 CONTINUE IF( NOUNIT ) \$ X( J ) = X( J )*A( J, J ) END IF 20 CONTINUE ELSE JX = KX DO 40, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = X( JX ) IX = KX DO 30, I = 1, J - 1 X( IX ) = X( IX ) + TEMP*A( I, J ) IX = IX + INCX 30 CONTINUE IF( NOUNIT ) \$ X( JX ) = X( JX )*A( J, J ) END IF JX = JX + INCX 40 CONTINUE END IF ELSE IF( INCX.EQ.1 )THEN DO 60, J = N, 1, -1 IF( X( J ).NE.ZERO )THEN TEMP = X( J ) DO 50, I = N, J + 1, -1 X( I ) = X( I ) + TEMP*A( I, J ) 50 CONTINUE IF( NOUNIT ) \$ X( J ) = X( J )*A( J, J ) END IF 60 CONTINUE ELSE KX = KX + ( N - 1 )*INCX JX = KX DO 80, J = N, 1, -1 IF( X( JX ).NE.ZERO )THEN TEMP = X( JX ) IX = KX DO 70, I = N, J + 1, -1 X( IX ) = X( IX ) + TEMP*A( I, J ) IX = IX - INCX 70 CONTINUE IF( NOUNIT ) \$ X( JX ) = X( JX )*A( J, J ) END IF JX = JX - INCX 80 CONTINUE END IF END IF ELSE * * Form x := A'*x. * IF( LSAME( UPLO, 'U' ) )THEN IF( INCX.EQ.1 )THEN DO 100, J = N, 1, -1 TEMP = X( J ) IF( NOUNIT ) \$ TEMP = TEMP*A( J, J ) DO 90, I = J - 1, 1, -1 TEMP = TEMP + A( I, J )*X( I ) 90 CONTINUE X( J ) = TEMP 100 CONTINUE ELSE JX = KX + ( N - 1 )*INCX DO 120, J = N, 1, -1 TEMP = X( JX ) IX = JX IF( NOUNIT ) \$ TEMP = TEMP*A( J, J ) DO 110, I = J - 1, 1, -1 IX = IX - INCX TEMP = TEMP + A( I, J )*X( IX ) 110 CONTINUE X( JX ) = TEMP JX = JX - INCX 120 CONTINUE END IF ELSE IF( INCX.EQ.1 )THEN DO 140, J = 1, N TEMP = X( J ) IF( NOUNIT ) \$ TEMP = TEMP*A( J, J ) DO 130, I = J + 1, N TEMP = TEMP + A( I, J )*X( I ) 130 CONTINUE X( J ) = TEMP 140 CONTINUE ELSE JX = KX DO 160, J = 1, N TEMP = X( JX ) IX = JX IF( NOUNIT ) \$ TEMP = TEMP*A( J, J ) DO 150, I = J + 1, N IX = IX + INCX TEMP = TEMP + A( I, J )*X( IX ) 150 CONTINUE X( JX ) = TEMP JX = JX + INCX 160 CONTINUE END IF END IF END IF * RETURN * * End of STRMV . * END SUBROUTINE STRSV ( UPLO, TRANS, DIAG, N, A, LDA, X, INCX ) * .. Scalar Arguments .. INTEGER INCX, LDA, N CHARACTER*1 DIAG, TRANS, UPLO * .. Array Arguments .. REAL A( LDA, * ), X( * ) * .. * * Purpose * ======= * * STRSV solves one of the systems of equations * * A*x = b, or A'*x = b, * * where b and x are n element vectors and A is an n by n unit, or * non-unit, upper or lower triangular matrix. * * No test for singularity or near-singularity is included in this * routine. Such tests must be performed before calling this routine. * * Parameters * ========== * * UPLO - CHARACTER*1. * On entry, UPLO specifies whether the matrix is an upper or * lower triangular matrix as follows: * * UPLO = 'U' or 'u' A is an upper triangular matrix. * * UPLO = 'L' or 'l' A is a lower triangular matrix. * * Unchanged on exit. * * TRANS - CHARACTER*1. * On entry, TRANS specifies the equations to be solved as * follows: * * TRANS = 'N' or 'n' A*x = b. * * TRANS = 'T' or 't' A'*x = b. * * TRANS = 'C' or 'c' A'*x = b. * * Unchanged on exit. * * DIAG - CHARACTER*1. * On entry, DIAG specifies whether or not A is unit * triangular as follows: * * DIAG = 'U' or 'u' A is assumed to be unit triangular. * * DIAG = 'N' or 'n' A is not assumed to be unit * triangular. * * Unchanged on exit. * * N - INTEGER. * On entry, N specifies the order of the matrix A. * N must be at least zero. * Unchanged on exit. * * A - REAL array of DIMENSION ( LDA, n ). * Before entry with UPLO = 'U' or 'u', the leading n by n * upper triangular part of the array A must contain the upper * triangular matrix and the strictly lower triangular part of * A is not referenced. * Before entry with UPLO = 'L' or 'l', the leading n by n * lower triangular part of the array A must contain the lower * triangular matrix and the strictly upper triangular part of * A is not referenced. * Note that when DIAG = 'U' or 'u', the diagonal elements of * A are not referenced either, but are assumed to be unity. * Unchanged on exit. * * LDA - INTEGER. * On entry, LDA specifies the first dimension of A as declared * in the calling (sub) program. LDA must be at least * max( 1, n ). * Unchanged on exit. * * X - REAL array of dimension at least * ( 1 + ( n - 1 )*abs( INCX ) ). * Before entry, the incremented array X must contain the n * element right-hand side vector b. On exit, X is overwritten * with the solution vector x. * * INCX - INTEGER. * On entry, INCX specifies the increment for the elements of * X. INCX must not be zero. * Unchanged on exit. * * * Level 2 Blas routine. * * -- Written on 22-October-1986. * Jack Dongarra, Argonne National Lab. * Jeremy Du Croz, Nag Central Office. * Sven Hammarling, Nag Central Office. * Richard Hanson, Sandia National Labs. * * * .. Parameters .. REAL ZERO PARAMETER ( ZERO = 0.0E+0 ) * .. Local Scalars .. REAL TEMP INTEGER I, INFO, IX, J, JX, KX LOGICAL NOUNIT * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. External Subroutines .. EXTERNAL XERBLA * .. Intrinsic Functions .. INTRINSIC MAX * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 IF ( .NOT.LSAME( UPLO , 'U' ).AND. \$ .NOT.LSAME( UPLO , 'L' ) )THEN INFO = 1 ELSE IF( .NOT.LSAME( TRANS, 'N' ).AND. \$ .NOT.LSAME( TRANS, 'T' ).AND. \$ .NOT.LSAME( TRANS, 'C' ) )THEN INFO = 2 ELSE IF( .NOT.LSAME( DIAG , 'U' ).AND. \$ .NOT.LSAME( DIAG , 'N' ) )THEN INFO = 3 ELSE IF( N.LT.0 )THEN INFO = 4 ELSE IF( LDA.LT.MAX( 1, N ) )THEN INFO = 6 ELSE IF( INCX.EQ.0 )THEN INFO = 8 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'STRSV ', INFO ) RETURN END IF * * Quick return if possible. * IF( N.EQ.0 ) \$ RETURN * NOUNIT = LSAME( DIAG, 'N' ) * * Set up the start point in X if the increment is not unity. This * will be ( N - 1 )*INCX too small for descending loops. * IF( INCX.LE.0 )THEN KX = 1 - ( N - 1 )*INCX ELSE IF( INCX.NE.1 )THEN KX = 1 END IF * * Start the operations. In this version the elements of A are * accessed sequentially with one pass through A. * IF( LSAME( TRANS, 'N' ) )THEN * * Form x := inv( A )*x. * IF( LSAME( UPLO, 'U' ) )THEN IF( INCX.EQ.1 )THEN DO 20, J = N, 1, -1 IF( X( J ).NE.ZERO )THEN IF( NOUNIT ) \$ X( J ) = X( J )/A( J, J ) TEMP = X( J ) DO 10, I = J - 1, 1, -1 X( I ) = X( I ) - TEMP*A( I, J ) 10 CONTINUE END IF 20 CONTINUE ELSE JX = KX + ( N - 1 )*INCX DO 40, J = N, 1, -1 IF( X( JX ).NE.ZERO )THEN IF( NOUNIT ) \$ X( JX ) = X( JX )/A( J, J ) TEMP = X( JX ) IX = JX DO 30, I = J - 1, 1, -1 IX = IX - INCX X( IX ) = X( IX ) - TEMP*A( I, J ) 30 CONTINUE END IF JX = JX - INCX 40 CONTINUE END IF ELSE IF( INCX.EQ.1 )THEN DO 60, J = 1, N IF( X( J ).NE.ZERO )THEN IF( NOUNIT ) \$ X( J ) = X( J )/A( J, J ) TEMP = X( J ) DO 50, I = J + 1, N X( I ) = X( I ) - TEMP*A( I, J ) 50 CONTINUE END IF 60 CONTINUE ELSE JX = KX DO 80, J = 1, N IF( X( JX ).NE.ZERO )THEN IF( NOUNIT ) \$ X( JX ) = X( JX )/A( J, J ) TEMP = X( JX ) IX = JX DO 70, I = J + 1, N IX = IX + INCX X( IX ) = X( IX ) - TEMP*A( I, J ) 70 CONTINUE END IF JX = JX + INCX 80 CONTINUE END IF END IF ELSE * * Form x := inv( A' )*x. * IF( LSAME( UPLO, 'U' ) )THEN IF( INCX.EQ.1 )THEN DO 100, J = 1, N TEMP = X( J ) DO 90, I = 1, J - 1 TEMP = TEMP - A( I, J )*X( I ) 90 CONTINUE IF( NOUNIT ) \$ TEMP = TEMP/A( J, J ) X( J ) = TEMP 100 CONTINUE ELSE JX = KX DO 120, J = 1, N TEMP = X( JX ) IX = KX DO 110, I = 1, J - 1 TEMP = TEMP - A( I, J )*X( IX ) IX = IX + INCX 110 CONTINUE IF( NOUNIT ) \$ TEMP = TEMP/A( J, J ) X( JX ) = TEMP JX = JX + INCX 120 CONTINUE END IF ELSE IF( INCX.EQ.1 )THEN DO 140, J = N, 1, -1 TEMP = X( J ) DO 130, I = N, J + 1, -1 TEMP = TEMP - A( I, J )*X( I ) 130 CONTINUE IF( NOUNIT ) \$ TEMP = TEMP/A( J, J ) X( J ) = TEMP 140 CONTINUE ELSE KX = KX + ( N - 1 )*INCX JX = KX DO 160, J = N, 1, -1 TEMP = X( JX ) IX = KX DO 150, I = N, J + 1, -1 TEMP = TEMP - A( I, J )*X( IX ) IX = IX - INCX 150 CONTINUE IF( NOUNIT ) \$ TEMP = TEMP/A( J, J ) X( JX ) = TEMP JX = JX - INCX 160 CONTINUE END IF END IF END IF * RETURN * * End of STRSV . * END SUBROUTINE SSYMV ( UPLO, N, ALPHA, A, LDA, X, INCX, \$ BETA, Y, INCY ) * .. Scalar Arguments .. REAL ALPHA, BETA INTEGER INCX, INCY, LDA, N CHARACTER*1 UPLO * .. Array Arguments .. REAL A( LDA, * ), X( * ), Y( * ) * .. * * Purpose * ======= * * SSYMV performs the matrix-vector operation * * y := alpha*A*x + beta*y, * * where alpha and beta are scalars, x and y are n element vectors and * A is an n by n symmetric matrix. * * Parameters * ========== * * UPLO - CHARACTER*1. * On entry, UPLO specifies whether the upper or lower * triangular part of the array A is to be referenced as * follows: * * UPLO = 'U' or 'u' Only the upper triangular part of A * is to be referenced. * * UPLO = 'L' or 'l' Only the lower triangular part of A * is to be referenced. * * Unchanged on exit. * * N - INTEGER. * On entry, N specifies the order of the matrix A. * N must be at least zero. * Unchanged on exit. * * ALPHA - REAL . * On entry, ALPHA specifies the scalar alpha. * Unchanged on exit. * * A - REAL array of DIMENSION ( LDA, n ). * Before entry with UPLO = 'U' or 'u', the leading n by n * upper triangular part of the array A must contain the upper * triangular part of the symmetric matrix and the strictly * lower triangular part of A is not referenced. * Before entry with UPLO = 'L' or 'l', the leading n by n * lower triangular part of the array A must contain the lower * triangular part of the symmetric matrix and the strictly * upper triangular part of A is not referenced. * Unchanged on exit. * * LDA - INTEGER. * On entry, LDA specifies the first dimension of A as declared * in the calling (sub) program. LDA must be at least * max( 1, n ). * Unchanged on exit. * * X - REAL array of dimension at least * ( 1 + ( n - 1 )*abs( INCX ) ). * Before entry, the incremented array X must contain the n * element vector x. * Unchanged on exit. * * INCX - INTEGER. * On entry, INCX specifies the increment for the elements of * X. INCX must not be zero. * Unchanged on exit. * * BETA - REAL . * On entry, BETA specifies the scalar beta. When BETA is * supplied as zero then Y need not be set on input. * Unchanged on exit. * * Y - REAL array of dimension at least * ( 1 + ( n - 1 )*abs( INCY ) ). * Before entry, the incremented array Y must contain the n * element vector y. On exit, Y is overwritten by the updated * vector y. * * INCY - INTEGER. * On entry, INCY specifies the increment for the elements of * Y. INCY must not be zero. * Unchanged on exit. * * * Level 2 Blas routine. * * -- Written on 22-October-1986. * Jack Dongarra, Argonne National Lab. * Jeremy Du Croz, Nag Central Office. * Sven Hammarling, Nag Central Office. * Richard Hanson, Sandia National Labs. * * * .. Parameters .. REAL ONE , ZERO PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) * .. Local Scalars .. REAL TEMP1, TEMP2 INTEGER I, INFO, IX, IY, J, JX, JY, KX, KY * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. External Subroutines .. EXTERNAL XERBLA * .. Intrinsic Functions .. INTRINSIC MAX * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 IF ( .NOT.LSAME( UPLO, 'U' ).AND. \$ .NOT.LSAME( UPLO, 'L' ) )THEN INFO = 1 ELSE IF( N.LT.0 )THEN INFO = 2 ELSE IF( LDA.LT.MAX( 1, N ) )THEN INFO = 5 ELSE IF( INCX.EQ.0 )THEN INFO = 7 ELSE IF( INCY.EQ.0 )THEN INFO = 10 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'SSYMV ', INFO ) RETURN END IF * * Quick return if possible. * IF( ( N.EQ.0 ).OR.( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) ) \$ RETURN * * Set up the start points in X and Y. * IF( INCX.GT.0 )THEN KX = 1 ELSE KX = 1 - ( N - 1 )*INCX END IF IF( INCY.GT.0 )THEN KY = 1 ELSE KY = 1 - ( N - 1 )*INCY END IF * * Start the operations. In this version the elements of A are * accessed sequentially with one pass through the triangular part * of A. * * First form y := beta*y. * IF( BETA.NE.ONE )THEN IF( INCY.EQ.1 )THEN IF( BETA.EQ.ZERO )THEN DO 10, I = 1, N Y( I ) = ZERO 10 CONTINUE ELSE DO 20, I = 1, N Y( I ) = BETA*Y( I ) 20 CONTINUE END IF ELSE IY = KY IF( BETA.EQ.ZERO )THEN DO 30, I = 1, N Y( IY ) = ZERO IY = IY + INCY 30 CONTINUE ELSE DO 40, I = 1, N Y( IY ) = BETA*Y( IY ) IY = IY + INCY 40 CONTINUE END IF END IF END IF IF( ALPHA.EQ.ZERO ) \$ RETURN IF( LSAME( UPLO, 'U' ) )THEN * * Form y when A is stored in upper triangle. * IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN DO 60, J = 1, N TEMP1 = ALPHA*X( J ) TEMP2 = ZERO DO 50, I = 1, J - 1 Y( I ) = Y( I ) + TEMP1*A( I, J ) TEMP2 = TEMP2 + A( I, J )*X( I ) 50 CONTINUE Y( J ) = Y( J ) + TEMP1*A( J, J ) + ALPHA*TEMP2 60 CONTINUE ELSE JX = KX JY = KY DO 80, J = 1, N TEMP1 = ALPHA*X( JX ) TEMP2 = ZERO IX = KX IY = KY DO 70, I = 1, J - 1 Y( IY ) = Y( IY ) + TEMP1*A( I, J ) TEMP2 = TEMP2 + A( I, J )*X( IX ) IX = IX + INCX IY = IY + INCY 70 CONTINUE Y( JY ) = Y( JY ) + TEMP1*A( J, J ) + ALPHA*TEMP2 JX = JX + INCX JY = JY + INCY 80 CONTINUE END IF ELSE * * Form y when A is stored in lower triangle. * IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN DO 100, J = 1, N TEMP1 = ALPHA*X( J ) TEMP2 = ZERO Y( J ) = Y( J ) + TEMP1*A( J, J ) DO 90, I = J + 1, N Y( I ) = Y( I ) + TEMP1*A( I, J ) TEMP2 = TEMP2 + A( I, J )*X( I ) 90 CONTINUE Y( J ) = Y( J ) + ALPHA*TEMP2 100 CONTINUE ELSE JX = KX JY = KY DO 120, J = 1, N TEMP1 = ALPHA*X( JX ) TEMP2 = ZERO Y( JY ) = Y( JY ) + TEMP1*A( J, J ) IX = JX IY = JY DO 110, I = J + 1, N IX = IX + INCX IY = IY + INCY Y( IY ) = Y( IY ) + TEMP1*A( I, J ) TEMP2 = TEMP2 + A( I, J )*X( IX ) 110 CONTINUE Y( JY ) = Y( JY ) + ALPHA*TEMP2 JX = JX + INCX JY = JY + INCY 120 CONTINUE END IF END IF * RETURN * * End of SSYMV . * END integer function isamax(n,sx,incx) c c finds the index of element having max. absolute value. c jack dongarra, linpack, 3/11/78. c modified 3/93 to return if incx .le. 0. c real sx(1),smax integer i,incx,ix,n c isamax = 0 if( n.lt.1 .or. incx.le.0 ) return isamax = 1 if(n.eq.1)return if(incx.eq.1)go to 20 c c code for increment not equal to 1 c ix = 1 smax = abs(sx(1)) ix = ix + incx do 10 i = 2,n if(abs(sx(ix)).le.smax) go to 5 isamax = i smax = abs(sx(ix)) 5 ix = ix + incx 10 continue return c c code for increment equal to 1 c 20 smax = abs(sx(1)) do 30 i = 2,n if(abs(sx(i)).le.smax) go to 30 isamax = i smax = abs(sx(i)) 30 continue return end SUBROUTINE XERBLA ( SRNAME, INFO ) * .. Scalar Arguments .. INTEGER INFO CHARACTER*6 SRNAME * .. * * Purpose * ======= * * XERBLA is an error handler for the Level 2 BLAS routines. * * It is called by the Level 2 BLAS routines if an input parameter is * invalid. * * Installers should consider modifying the STOP statement in order to * call system-specific exception-handling facilities. * * Parameters * ========== * * SRNAME - CHARACTER*6. * On entry, SRNAME specifies the name of the routine which * called XERBLA. * * INFO - INTEGER. * On entry, INFO specifies the position of the invalid * parameter in the parameter-list of the calling routine. * * * Auxiliary routine for Level 2 Blas. * * Written on 20-July-1986. * * .. Executable Statements .. * WRITE (*,99999) SRNAME, INFO * STOP * 99999 FORMAT ( ' ** On entry to ', A6, ' parameter number ', I2, \$ ' had an illegal value' ) * * End of XERBLA. * END LOGICAL FUNCTION LSAME( CA, CB ) * * -- LAPACK auxiliary routine (version 1.1) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * February 29, 1992 * * .. Scalar Arguments .. CHARACTER CA, CB * .. * * Purpose * ======= * * LSAME returns .TRUE. if CA is the same letter as CB regardless of * case. * * Arguments * ========= * * CA (input) CHARACTER*1 * CB (input) CHARACTER*1 * CA and CB specify the single characters to be compared. * * .. Intrinsic Functions .. INTRINSIC ICHAR * .. * .. Local Scalars .. INTEGER INTA, INTB, ZCODE * .. * .. Executable Statements .. * * Test if the characters are equal * LSAME = CA.EQ.CB IF( LSAME ) \$ RETURN * * Now test for equivalence if both characters are alphabetic. * ZCODE = ICHAR( 'Z' ) * * Use 'Z' rather than 'A' so that ASCII can be detected on Prime * machines, on which ICHAR returns a value with bit 8 set. * ICHAR('A') on Prime machines returns 193 which is the same as * ICHAR('A') on an EBCDIC machine. * INTA = ICHAR( CA ) INTB = ICHAR( CB ) * IF( ZCODE.EQ.90 .OR. ZCODE.EQ.122 ) THEN * * ASCII is assumed - ZCODE is the ASCII code of either lower or * upper case 'Z'. * IF( INTA.GE.97 .AND. INTA.LE.122 ) INTA = INTA - 32 IF( INTB.GE.97 .AND. INTB.LE.122 ) INTB = INTB - 32 * ELSE IF( ZCODE.EQ.233 .OR. ZCODE.EQ.169 ) THEN * * EBCDIC is assumed - ZCODE is the EBCDIC code of either lower or * upper case 'Z'. * IF( INTA.GE.129 .AND. INTA.LE.137 .OR. \$ INTA.GE.145 .AND. INTA.LE.153 .OR. \$ INTA.GE.162 .AND. INTA.LE.169 ) INTA = INTA + 64 IF( INTB.GE.129 .AND. INTB.LE.137 .OR. \$ INTB.GE.145 .AND. INTB.LE.153 .OR. \$ INTB.GE.162 .AND. INTB.LE.169 ) INTB = INTB + 64 * ELSE IF( ZCODE.EQ.218 .OR. ZCODE.EQ.250 ) THEN * * ASCII is assumed, on Prime machines - ZCODE is the ASCII code * plus 128 of either lower or upper case 'Z'. * IF( INTA.GE.225 .AND. INTA.LE.250 ) INTA = INTA - 32 IF( INTB.GE.225 .AND. INTB.LE.250 ) INTB = INTB - 32 END IF LSAME = INTA.EQ.INTB * * RETURN * * End of LSAME * END