00001 SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO ) 00002 * 00003 * -- LAPACK routine (version 3.2) -- 00004 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00005 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00006 * November 2006 00007 * 00008 * .. Scalar Arguments .. 00009 CHARACTER UPLO 00010 INTEGER INFO, N 00011 * .. 00012 * .. Array Arguments .. 00013 DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * ) 00014 * .. 00015 * 00016 * Purpose 00017 * ======= 00018 * 00019 * DSPTRD reduces a real symmetric matrix A stored in packed form to 00020 * symmetric tridiagonal form T by an orthogonal similarity 00021 * transformation: Q**T * A * Q = T. 00022 * 00023 * Arguments 00024 * ========= 00025 * 00026 * UPLO (input) CHARACTER*1 00027 * = 'U': Upper triangle of A is stored; 00028 * = 'L': Lower triangle of A is stored. 00029 * 00030 * N (input) INTEGER 00031 * The order of the matrix A. N >= 0. 00032 * 00033 * AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) 00034 * On entry, the upper or lower triangle of the symmetric matrix 00035 * A, packed columnwise in a linear array. The j-th column of A 00036 * is stored in the array AP as follows: 00037 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 00038 * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. 00039 * On exit, if UPLO = 'U', the diagonal and first superdiagonal 00040 * of A are overwritten by the corresponding elements of the 00041 * tridiagonal matrix T, and the elements above the first 00042 * superdiagonal, with the array TAU, represent the orthogonal 00043 * matrix Q as a product of elementary reflectors; if UPLO 00044 * = 'L', the diagonal and first subdiagonal of A are over- 00045 * written by the corresponding elements of the tridiagonal 00046 * matrix T, and the elements below the first subdiagonal, with 00047 * the array TAU, represent the orthogonal matrix Q as a product 00048 * of elementary reflectors. See Further Details. 00049 * 00050 * D (output) DOUBLE PRECISION array, dimension (N) 00051 * The diagonal elements of the tridiagonal matrix T: 00052 * D(i) = A(i,i). 00053 * 00054 * E (output) DOUBLE PRECISION array, dimension (N-1) 00055 * The off-diagonal elements of the tridiagonal matrix T: 00056 * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. 00057 * 00058 * TAU (output) DOUBLE PRECISION array, dimension (N-1) 00059 * The scalar factors of the elementary reflectors (see Further 00060 * Details). 00061 * 00062 * INFO (output) INTEGER 00063 * = 0: successful exit 00064 * < 0: if INFO = -i, the i-th argument had an illegal value 00065 * 00066 * Further Details 00067 * =============== 00068 * 00069 * If UPLO = 'U', the matrix Q is represented as a product of elementary 00070 * reflectors 00071 * 00072 * Q = H(n-1) . . . H(2) H(1). 00073 * 00074 * Each H(i) has the form 00075 * 00076 * H(i) = I - tau * v * v' 00077 * 00078 * where tau is a real scalar, and v is a real vector with 00079 * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP, 00080 * overwriting A(1:i-1,i+1), and tau is stored in TAU(i). 00081 * 00082 * If UPLO = 'L', the matrix Q is represented as a product of elementary 00083 * reflectors 00084 * 00085 * Q = H(1) H(2) . . . H(n-1). 00086 * 00087 * Each H(i) has the form 00088 * 00089 * H(i) = I - tau * v * v' 00090 * 00091 * where tau is a real scalar, and v is a real vector with 00092 * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP, 00093 * overwriting A(i+2:n,i), and tau is stored in TAU(i). 00094 * 00095 * ===================================================================== 00096 * 00097 * .. Parameters .. 00098 DOUBLE PRECISION ONE, ZERO, HALF 00099 PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0, 00100 $ HALF = 1.0D0 / 2.0D0 ) 00101 * .. 00102 * .. Local Scalars .. 00103 LOGICAL UPPER 00104 INTEGER I, I1, I1I1, II 00105 DOUBLE PRECISION ALPHA, TAUI 00106 * .. 00107 * .. External Subroutines .. 00108 EXTERNAL DAXPY, DLARFG, DSPMV, DSPR2, XERBLA 00109 * .. 00110 * .. External Functions .. 00111 LOGICAL LSAME 00112 DOUBLE PRECISION DDOT 00113 EXTERNAL LSAME, DDOT 00114 * .. 00115 * .. Executable Statements .. 00116 * 00117 * Test the input parameters 00118 * 00119 INFO = 0 00120 UPPER = LSAME( UPLO, 'U' ) 00121 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00122 INFO = -1 00123 ELSE IF( N.LT.0 ) THEN 00124 INFO = -2 00125 END IF 00126 IF( INFO.NE.0 ) THEN 00127 CALL XERBLA( 'DSPTRD', -INFO ) 00128 RETURN 00129 END IF 00130 * 00131 * Quick return if possible 00132 * 00133 IF( N.LE.0 ) 00134 $ RETURN 00135 * 00136 IF( UPPER ) THEN 00137 * 00138 * Reduce the upper triangle of A. 00139 * I1 is the index in AP of A(1,I+1). 00140 * 00141 I1 = N*( N-1 ) / 2 + 1 00142 DO 10 I = N - 1, 1, -1 00143 * 00144 * Generate elementary reflector H(i) = I - tau * v * v' 00145 * to annihilate A(1:i-1,i+1) 00146 * 00147 CALL DLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI ) 00148 E( I ) = AP( I1+I-1 ) 00149 * 00150 IF( TAUI.NE.ZERO ) THEN 00151 * 00152 * Apply H(i) from both sides to A(1:i,1:i) 00153 * 00154 AP( I1+I-1 ) = ONE 00155 * 00156 * Compute y := tau * A * v storing y in TAU(1:i) 00157 * 00158 CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU, 00159 $ 1 ) 00160 * 00161 * Compute w := y - 1/2 * tau * (y'*v) * v 00162 * 00163 ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 ) 00164 CALL DAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 ) 00165 * 00166 * Apply the transformation as a rank-2 update: 00167 * A := A - v * w' - w * v' 00168 * 00169 CALL DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP ) 00170 * 00171 AP( I1+I-1 ) = E( I ) 00172 END IF 00173 D( I+1 ) = AP( I1+I ) 00174 TAU( I ) = TAUI 00175 I1 = I1 - I 00176 10 CONTINUE 00177 D( 1 ) = AP( 1 ) 00178 ELSE 00179 * 00180 * Reduce the lower triangle of A. II is the index in AP of 00181 * A(i,i) and I1I1 is the index of A(i+1,i+1). 00182 * 00183 II = 1 00184 DO 20 I = 1, N - 1 00185 I1I1 = II + N - I + 1 00186 * 00187 * Generate elementary reflector H(i) = I - tau * v * v' 00188 * to annihilate A(i+2:n,i) 00189 * 00190 CALL DLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI ) 00191 E( I ) = AP( II+1 ) 00192 * 00193 IF( TAUI.NE.ZERO ) THEN 00194 * 00195 * Apply H(i) from both sides to A(i+1:n,i+1:n) 00196 * 00197 AP( II+1 ) = ONE 00198 * 00199 * Compute y := tau * A * v storing y in TAU(i:n-1) 00200 * 00201 CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1, 00202 $ ZERO, TAU( I ), 1 ) 00203 * 00204 * Compute w := y - 1/2 * tau * (y'*v) * v 00205 * 00206 ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ), 00207 $ 1 ) 00208 CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 ) 00209 * 00210 * Apply the transformation as a rank-2 update: 00211 * A := A - v * w' - w * v' 00212 * 00213 CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1, 00214 $ AP( I1I1 ) ) 00215 * 00216 AP( II+1 ) = E( I ) 00217 END IF 00218 D( I ) = AP( II ) 00219 TAU( I ) = TAUI 00220 II = I1I1 00221 20 CONTINUE 00222 D( N ) = AP( II ) 00223 END IF 00224 * 00225 RETURN 00226 * 00227 * End of DSPTRD 00228 * 00229 END
1.7.2