1*> \brief \b SSPTRD 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download SSPTRD + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ssptrd.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ssptrd.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ssptrd.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE SSPTRD( UPLO, N, AP, D, E, TAU, INFO ) 22* 23* .. Scalar Arguments .. 24* CHARACTER UPLO 25* INTEGER INFO, N 26* .. 27* .. Array Arguments .. 28* REAL AP( * ), D( * ), E( * ), TAU( * ) 29* .. 30* 31* 32*> \par Purpose: 33* ============= 34*> 35*> \verbatim 36*> 37*> SSPTRD reduces a real symmetric matrix A stored in packed form to 38*> symmetric tridiagonal form T by an orthogonal similarity 39*> transformation: Q**T * A * Q = T. 40*> \endverbatim 41* 42* Arguments: 43* ========== 44* 45*> \param[in] UPLO 46*> \verbatim 47*> UPLO is CHARACTER*1 48*> = 'U': Upper triangle of A is stored; 49*> = 'L': Lower triangle of A is stored. 50*> \endverbatim 51*> 52*> \param[in] N 53*> \verbatim 54*> N is INTEGER 55*> The order of the matrix A. N >= 0. 56*> \endverbatim 57*> 58*> \param[in,out] AP 59*> \verbatim 60*> AP is REAL array, dimension (N*(N+1)/2) 61*> On entry, the upper or lower triangle of the symmetric matrix 62*> A, packed columnwise in a linear array. The j-th column of A 63*> is stored in the array AP as follows: 64*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 65*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. 66*> On exit, if UPLO = 'U', the diagonal and first superdiagonal 67*> of A are overwritten by the corresponding elements of the 68*> tridiagonal matrix T, and the elements above the first 69*> superdiagonal, with the array TAU, represent the orthogonal 70*> matrix Q as a product of elementary reflectors; if UPLO 71*> = 'L', the diagonal and first subdiagonal of A are over- 72*> written by the corresponding elements of the tridiagonal 73*> matrix T, and the elements below the first subdiagonal, with 74*> the array TAU, represent the orthogonal matrix Q as a product 75*> of elementary reflectors. See Further Details. 76*> \endverbatim 77*> 78*> \param[out] D 79*> \verbatim 80*> D is REAL array, dimension (N) 81*> The diagonal elements of the tridiagonal matrix T: 82*> D(i) = A(i,i). 83*> \endverbatim 84*> 85*> \param[out] E 86*> \verbatim 87*> E is REAL array, dimension (N-1) 88*> The off-diagonal elements of the tridiagonal matrix T: 89*> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. 90*> \endverbatim 91*> 92*> \param[out] TAU 93*> \verbatim 94*> TAU is REAL array, dimension (N-1) 95*> The scalar factors of the elementary reflectors (see Further 96*> Details). 97*> \endverbatim 98*> 99*> \param[out] INFO 100*> \verbatim 101*> INFO is INTEGER 102*> = 0: successful exit 103*> < 0: if INFO = -i, the i-th argument had an illegal value 104*> \endverbatim 105* 106* Authors: 107* ======== 108* 109*> \author Univ. of Tennessee 110*> \author Univ. of California Berkeley 111*> \author Univ. of Colorado Denver 112*> \author NAG Ltd. 113* 114*> \date November 2011 115* 116*> \ingroup realOTHERcomputational 117* 118*> \par Further Details: 119* ===================== 120*> 121*> \verbatim 122*> 123*> If UPLO = 'U', the matrix Q is represented as a product of elementary 124*> reflectors 125*> 126*> Q = H(n-1) . . . H(2) H(1). 127*> 128*> Each H(i) has the form 129*> 130*> H(i) = I - tau * v * v**T 131*> 132*> where tau is a real scalar, and v is a real vector with 133*> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP, 134*> overwriting A(1:i-1,i+1), and tau is stored in TAU(i). 135*> 136*> If UPLO = 'L', the matrix Q is represented as a product of elementary 137*> reflectors 138*> 139*> Q = H(1) H(2) . . . H(n-1). 140*> 141*> Each H(i) has the form 142*> 143*> H(i) = I - tau * v * v**T 144*> 145*> where tau is a real scalar, and v is a real vector with 146*> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP, 147*> overwriting A(i+2:n,i), and tau is stored in TAU(i). 148*> \endverbatim 149*> 150* ===================================================================== 151 SUBROUTINE SSPTRD( UPLO, N, AP, D, E, TAU, INFO ) 152* 153* -- LAPACK computational routine (version 3.4.0) -- 154* -- LAPACK is a software package provided by Univ. of Tennessee, -- 155* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 156* November 2011 157* 158* .. Scalar Arguments .. 159 CHARACTER UPLO 160 INTEGER INFO, N 161* .. 162* .. Array Arguments .. 163 REAL AP( * ), D( * ), E( * ), TAU( * ) 164* .. 165* 166* ===================================================================== 167* 168* .. Parameters .. 169 REAL ONE, ZERO, HALF 170 PARAMETER ( ONE = 1.0, ZERO = 0.0, HALF = 1.0 / 2.0 ) 171* .. 172* .. Local Scalars .. 173 LOGICAL UPPER 174 INTEGER I, I1, I1I1, II 175 REAL ALPHA, TAUI 176* .. 177* .. External Subroutines .. 178 EXTERNAL SAXPY, SLARFG, SSPMV, SSPR2, XERBLA 179* .. 180* .. External Functions .. 181 LOGICAL LSAME 182 REAL SDOT 183 EXTERNAL LSAME, SDOT 184* .. 185* .. Executable Statements .. 186* 187* Test the input parameters 188* 189 INFO = 0 190 UPPER = LSAME( UPLO, 'U' ) 191 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 192 INFO = -1 193 ELSE IF( N.LT.0 ) THEN 194 INFO = -2 195 END IF 196 IF( INFO.NE.0 ) THEN 197 CALL XERBLA( 'SSPTRD', -INFO ) 198 RETURN 199 END IF 200* 201* Quick return if possible 202* 203 IF( N.LE.0 ) 204 $ RETURN 205* 206 IF( UPPER ) THEN 207* 208* Reduce the upper triangle of A. 209* I1 is the index in AP of A(1,I+1). 210* 211 I1 = N*( N-1 ) / 2 + 1 212 DO 10 I = N - 1, 1, -1 213* 214* Generate elementary reflector H(i) = I - tau * v * v**T 215* to annihilate A(1:i-1,i+1) 216* 217 CALL SLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI ) 218 E( I ) = AP( I1+I-1 ) 219* 220 IF( TAUI.NE.ZERO ) THEN 221* 222* Apply H(i) from both sides to A(1:i,1:i) 223* 224 AP( I1+I-1 ) = ONE 225* 226* Compute y := tau * A * v storing y in TAU(1:i) 227* 228 CALL SSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU, 229 $ 1 ) 230* 231* Compute w := y - 1/2 * tau * (y**T *v) * v 232* 233 ALPHA = -HALF*TAUI*SDOT( I, TAU, 1, AP( I1 ), 1 ) 234 CALL SAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 ) 235* 236* Apply the transformation as a rank-2 update: 237* A := A - v * w**T - w * v**T 238* 239 CALL SSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP ) 240* 241 AP( I1+I-1 ) = E( I ) 242 END IF 243 D( I+1 ) = AP( I1+I ) 244 TAU( I ) = TAUI 245 I1 = I1 - I 246 10 CONTINUE 247 D( 1 ) = AP( 1 ) 248 ELSE 249* 250* Reduce the lower triangle of A. II is the index in AP of 251* A(i,i) and I1I1 is the index of A(i+1,i+1). 252* 253 II = 1 254 DO 20 I = 1, N - 1 255 I1I1 = II + N - I + 1 256* 257* Generate elementary reflector H(i) = I - tau * v * v**T 258* to annihilate A(i+2:n,i) 259* 260 CALL SLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI ) 261 E( I ) = AP( II+1 ) 262* 263 IF( TAUI.NE.ZERO ) THEN 264* 265* Apply H(i) from both sides to A(i+1:n,i+1:n) 266* 267 AP( II+1 ) = ONE 268* 269* Compute y := tau * A * v storing y in TAU(i:n-1) 270* 271 CALL SSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1, 272 $ ZERO, TAU( I ), 1 ) 273* 274* Compute w := y - 1/2 * tau * (y**T *v) * v 275* 276 ALPHA = -HALF*TAUI*SDOT( N-I, TAU( I ), 1, AP( II+1 ), 277 $ 1 ) 278 CALL SAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 ) 279* 280* Apply the transformation as a rank-2 update: 281* A := A - v * w**T - w * v**T 282* 283 CALL SSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1, 284 $ AP( I1I1 ) ) 285* 286 AP( II+1 ) = E( I ) 287 END IF 288 D( I ) = AP( II ) 289 TAU( I ) = TAUI 290 II = I1I1 291 20 CONTINUE 292 D( N ) = AP( II ) 293 END IF 294* 295 RETURN 296* 297* End of SSPTRD 298* 299 END 300