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SUBROUTINE ORTHES(NM,N,LOW,IGH,A,ORT)
C
INTEGER I,J,M,N,II,JJ,LA,MP,NM,IGH,KP1,LOW
DOUBLE PRECISION A(NM,N),ORT(IGH)
DOUBLE PRECISION F,G,H,SCALE
C
C THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE ORTHES,
C NUM. MATH. 12, 349-368(1968) BY MARTIN AND WILKINSON.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971).
C
C GIVEN A REAL GENERAL MATRIX, THIS SUBROUTINE
C REDUCES A SUBMATRIX SITUATED IN ROWS AND COLUMNS
C LOW THROUGH IGH TO UPPER HESSENBERG FORM BY
C ORTHOGONAL SIMILARITY TRANSFORMATIONS.
C
C ON INPUT
C
C NM MUST BE SET TO THE ROW DIMENSION OF TWO-DIMENSIONAL
C ARRAY PARAMETERS AS DECLARED IN THE CALLING PROGRAM
C DIMENSION STATEMENT.
C
C N IS THE ORDER OF THE MATRIX.
C
C LOW AND IGH ARE INTEGERS DETERMINED BY THE BALANCING
C SUBROUTINE BALANC. IF BALANC HAS NOT BEEN USED,
C SET LOW=1, IGH=N.
C
C A CONTAINS THE INPUT MATRIX.
C
C ON OUTPUT
C
C A CONTAINS THE HESSENBERG MATRIX. INFORMATION ABOUT
C THE ORTHOGONAL TRANSFORMATIONS USED IN THE REDUCTION
C IS STORED IN THE REMAINING TRIANGLE UNDER THE
C HESSENBERG MATRIX.
C
C ORT CONTAINS FURTHER INFORMATION ABOUT THE TRANSFORMATIONS.
C ONLY ELEMENTS LOW THROUGH IGH ARE USED.
C
C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C THIS VERSION DATED AUGUST 1983.
C
C ------------------------------------------------------------------
C
LA = IGH - 1
KP1 = LOW + 1
IF (LA .LT. KP1) GO TO 200
C
DO 180 M = KP1, LA
H = 0.0D0
ORT(M) = 0.0D0
SCALE = 0.0D0
C .......... SCALE COLUMN (ALGOL TOL THEN NOT NEEDED) ..........
DO 90 I = M, IGH
90 SCALE = SCALE + DABS(A(I,M-1))
C
IF (SCALE .EQ. 0.0D0) GO TO 180
MP = M + IGH
C .......... FOR I=IGH STEP -1 UNTIL M DO -- ..........
DO 100 II = M, IGH
I = MP - II
ORT(I) = A(I,M-1) / SCALE
H = H + ORT(I) * ORT(I)
100 CONTINUE
C
G = -DSIGN(DSQRT(H),ORT(M))
H = H - ORT(M) * G
ORT(M) = ORT(M) - G
C .......... FORM (I-(U*UT)/H) * A ..........
DO 130 J = M, N
F = 0.0D0
C .......... FOR I=IGH STEP -1 UNTIL M DO -- ..........
DO 110 II = M, IGH
I = MP - II
F = F + ORT(I) * A(I,J)
110 CONTINUE
C
F = F / H
C
DO 120 I = M, IGH
120 A(I,J) = A(I,J) - F * ORT(I)
C
130 CONTINUE
C .......... FORM (I-(U*UT)/H)*A*(I-(U*UT)/H) ..........
DO 160 I = 1, IGH
F = 0.0D0
C .......... FOR J=IGH STEP -1 UNTIL M DO -- ..........
DO 140 JJ = M, IGH
J = MP - JJ
F = F + ORT(J) * A(I,J)
140 CONTINUE
C
F = F / H
C
DO 150 J = M, IGH
150 A(I,J) = A(I,J) - F * ORT(J)
C
160 CONTINUE
C
ORT(M) = SCALE * ORT(M)
A(M,M-1) = SCALE * G
180 CONTINUE
C
200 RETURN
END
SUBROUTINE TRED1(NM,N,A,D,E,E2)
C
INTEGER I,J,K,L,N,II,NM,JP1
DOUBLE PRECISION A(NM,N),D(N),E(N),E2(N)
DOUBLE PRECISION F,G,H,SCALE
C
C THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE TRED1,
C NUM. MATH. 11, 181-195(1968) BY MARTIN, REINSCH, AND WILKINSON.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
C
C THIS SUBROUTINE REDUCES A REAL SYMMETRIC MATRIX
C TO A SYMMETRIC TRIDIAGONAL MATRIX USING
C ORTHOGONAL SIMILARITY TRANSFORMATIONS.
C
C ON INPUT
C
C NM MUST BE SET TO THE ROW DIMENSION OF TWO-DIMENSIONAL
C ARRAY PARAMETERS AS DECLARED IN THE CALLING PROGRAM
C DIMENSION STATEMENT.
C
C N IS THE ORDER OF THE MATRIX.
C
C A CONTAINS THE REAL SYMMETRIC INPUT MATRIX. ONLY THE
C LOWER TRIANGLE OF THE MATRIX NEED BE SUPPLIED.
C
C ON OUTPUT
C
C A CONTAINS INFORMATION ABOUT THE ORTHOGONAL TRANS-
C FORMATIONS USED IN THE REDUCTION IN ITS STRICT LOWER
C TRIANGLE. THE FULL UPPER TRIANGLE OF A IS UNALTERED.
C
C D CONTAINS THE DIAGONAL ELEMENTS OF THE TRIDIAGONAL MATRIX.
C
C E CONTAINS THE SUBDIAGONAL ELEMENTS OF THE TRIDIAGONAL
C MATRIX IN ITS LAST N-1 POSITIONS. E(1) IS SET TO ZERO.
C
C E2 CONTAINS THE SQUARES OF THE CORRESPONDING ELEMENTS OF E.
C E2 MAY COINCIDE WITH E IF THE SQUARES ARE NOT NEEDED.
C
C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C THIS VERSION DATED AUGUST 1983.
C
C ------------------------------------------------------------------
C
DO 100 I = 1, N
D(I) = A(N,I)
A(N,I) = A(I,I)
100 CONTINUE
C .......... FOR I=N STEP -1 UNTIL 1 DO -- ..........
DO 300 II = 1, N
I = N + 1 - II
L = I - 1
H = 0.0D0
SCALE = 0.0D0
IF (L .LT. 1) GO TO 130
C .......... SCALE ROW (ALGOL TOL THEN NOT NEEDED) ..........
DO 120 K = 1, L
120 SCALE = SCALE + DABS(D(K))
C
IF (SCALE .NE. 0.0D0) GO TO 140
C
DO 125 J = 1, L
D(J) = A(L,J)
A(L,J) = A(I,J)
A(I,J) = 0.0D0
125 CONTINUE
C
130 E(I) = 0.0D0
E2(I) = 0.0D0
GO TO 300
C
140 DO 150 K = 1, L
D(K) = D(K) / SCALE
H = H + D(K) * D(K)
150 CONTINUE
C
E2(I) = SCALE * SCALE * H
F = D(L)
G = -DSIGN(DSQRT(H),F)
E(I) = SCALE * G
H = H - F * G
D(L) = F - G
IF (L .EQ. 1) GO TO 285
C .......... FORM A*U ..........
DO 170 J = 1, L
170 E(J) = 0.0D0
C
DO 240 J = 1, L
F = D(J)
G = E(J) + A(J,J) * F
JP1 = J + 1
IF (L .LT. JP1) GO TO 220
C
DO 200 K = JP1, L
G = G + A(K,J) * D(K)
E(K) = E(K) + A(K,J) * F
200 CONTINUE
C
220 E(J) = G
240 CONTINUE
C .......... FORM P ..........
F = 0.0D0
C
DO 245 J = 1, L
E(J) = E(J) / H
F = F + E(J) * D(J)
245 CONTINUE
C
H = F / (H + H)
C .......... FORM Q ..........
DO 250 J = 1, L
250 E(J) = E(J) - H * D(J)
C .......... FORM REDUCED A ..........
DO 280 J = 1, L
F = D(J)
G = E(J)
C
DO 260 K = J, L
260 A(K,J) = A(K,J) - F * E(K) - G * D(K)
C
280 CONTINUE
C
285 DO 290 J = 1, L
F = D(J)
D(J) = A(L,J)
A(L,J) = A(I,J)
A(I,J) = F * SCALE
290 CONTINUE
C
300 CONTINUE
C
RETURN
END
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