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How to unite intervals?

Help me please to unite resulted intervals!

The line "For..." outputs the intervals where the roots exist (roots: 2.94 and 5,52). I have to consider a remark:

If in the intervals {x~[i],x[i+1]} and {x[i+1],x~~[i+1]} can be the roots of the equation, the range {x~[i],x~~[i+1]} must have at least one of its root.

X = {-2, 6}
        spx = {-2, -1.90577, -1.81153, -1.59327, -1.375, -1.35785, -1.3407,    -1.24655, -1.22941, -1.11811, -0.934054, -0.80167, -0.75, -0.625,-0.5, -0.25, -0.0981238, 0.303752, 0.651876, 0.94833, 1, 1.5, 1.75,2.11731, 2.5, 2.5625, 2.625, 3.3125, 3.75, 4, 4.00964, 4.01928,4.25964, 4.36731, 4.5, 4.75, 5, 5.25, 5.5, 5.75, 6}
        spfw = {33.3632, 43.263, 51.6709, 55.5421, 57.1266, 57.2511, 57.3756,58.059, 58.0778, 58.1995,56.846,55.1903,54.5739,53.0828,51.1542,48.9959,48.0325,42.2533, 36.408,30.7952,30.1551,28.6446,23.138,19.4168,6.47053,5.90328,5.32951,-0.513959, -0.750527, -6.38895, -6.39157, -6.39418,-6.36456, -6.09357, -6.28599, -5.25369, -4.19539, -2.18625, -0.133803,2.90414, 6.171}
        spfn = {33.3632, 40.2933, 46.5882, 51.9781, 55.5583, 55.5708, 55.5762, 55.4604, 55.4393, 55.0045, 530116, 51.1309, 50.4546, 48.1226, 45.6012, 43.402, 42.066, 37.5522, 32.6864, 28.1979, 28.0685,25.7067, 17.5943,13.5547, -2.97428, -3.21054, -3.36422, -5.05466, -5.1301, -6.4392,-6.76879, -6.48231, -7.20196, -7.00719, -7.53373, -6.00246, -4.41058,-2.8187, -1.16621, 2.35765, 6.04694}

        For[i = 1, i < Length@spfn, i++,
         If[! (((0 < spfn[[i]]) && (0 < spfn[[i + 1]])) || 
           ((spfw[[i]] < 0) && (spfw[[i + 1]] < 0))),
           Print["1) exists root on: {", spx[[i]], ";", spx[[i + 1]], "}"]]]

So the result include 5 intervals:

1) exists root on: {2.11731;2.5}
1) exists root on: {2.5;2.5625}
1) exists root on: {2.5625;2.625}
1) exists root on: {2.625;3.3125}
1) exists root on: {5.5;5.75}

And as the first root is 2.94 it has to enter into 4 first intervals, and 5.52 in the last. So after considering the remark, that line should outputs two intervals.I tried to use IntervalUnion but it doesn't work:( Please help me to code this remark in this line.
POSTED BY: Julia Ilkiv
8 Replies
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Bill Simpson
Bill Simpson
Posted 9 years ago

Can you show the exact code and input you used with IntervalUnion that did not work? Did your code look exactly like the examples shown on the help page for IntervalUnion?
POSTED BY: Bill Simpson

Yes, but it outputs 5 intervals and union nothing:( 

 For[i = 1, i < Length@spfn, i++,
     If[! (((0 < spfn[[i]]) && (0 < spfn[[i + 1]])) || ((spfw[[i]] < 
             0) && (spfw[[i + 1]] < 0))),
      Print[IntervalUnion[Interval[{spx[[i]], spx[[i + 1]]}], 
        Interval[{spx[[i]], spx[[i + 1]]}]]]
      ]
     ]
POSTED BY: Julia Ilkiv

Help me, please :(
POSTED BY: Julia Ilkiv
Bill Simpson
Bill Simpson
Posted 9 years ago
POSTED BY: Bill Simpson

Thank you very much!!! You absolutely helped me:)
POSTED BY: Julia Ilkiv

I'm sorry for bothering..... It was the 1-st step of my algorithm. Tell me please, how on the 2-nd step of algorithm to find intervals roots separately in the 1-st found interval {2.11731,3.3125} and in the second {5.5,5.75}, using myintervals1 ???? I tried this, but I can't correctly use it ;((

X = {-2, 6}
spx = {-2, -1.90577, -1.81153, -1.59327, -1.375, -1.35785, -1.3407,  -1.24655, -1.22941, -1.11811, -0.934054, -0.80167, -0.75, -0.625,  -0.5, -0.25, -0.0981238, 0.303752, 0.651876, 0.94833, 1, 1.5, 1.75, 2.11731, 2.5, 2.5625, 2.625, 3.3125, 3.75, 4, 4.00964, 4.01928,  4.25964, 4.36731, 4.5, 4.75, 5, 5.25, 5.5, 5.75, 6}
spkw = {105.056, 89.2249, 17.7361, 7.25929, 7.25929, 7.25929, 7.25929, 1.09386,  1.09386, -7.35382, -12.5073, -11.929, -11.929, -15.429, -8.63312, -6.34314, -14.3807, -16.7907, -18.933, -12.3896, -3.021, -22.0262,  -25.7865, -18.8033, -9.07591, -9.18036, -8.49959, -9.24378, -7.32337,  -0.271835, -0.270096, 0.123206, 0.156523, 0.465142, 4.12922, 4.23318, 
  8.03654, 8.20981, 12.1518, 12.3944}
spkn = {73.5426, 66.8007, 24.6942, 16.4029, 0.726929,  0.314512, -1.23002, -1.23002, -3.90668, -10.8276, -14.2065,     -13.0895, -18.656, -20.1709, -8.79676, -8.79676, -11.2319, -13.9771, -15.1407, -2.50312, -4.72374, -32.4496, -34.2958, -21.0455, -2.45882,  -2.45882, -2.45882, -2.45882, -2.45882, -2.45882, -2.45882, -1.70357,  -1.70357, -1.11799, 6.1251, 6.36752, 6.36752, 6.60995, 14.0955,  14.5803}
spmw = {243.475, 213.305, 83.8004, 67.1081, 67.1081, 67.1081, 67.1081,  59.4226, 59.4226, 49.9772, 45.1635, 45.6272, 45.6272, 43.4397,   46.8376, 47.4101, 46.6214, 47.3535, 48.75, 42.5447, 33.1761,  61.6839, 68.2644, 53.4787, 29.1603, 29.4279, 27.6409, 30.1061,  22.9045, -5.30161, -5.30859, -6.88938, -7.0313, -8.37913, -24.8675,  -25.3613, -44.3781, -45.2877, -66.9686, -68.3634}
spmn = {180.448, 167.6, 91.3225, 78.1123, 56.5579, 55.9978, 53.9271,  53.9271, 50.6364, 42.898, 39.742, 40.6374, 36.4626, 35.5158,    41.2028, 41.2028, 40.9639, 41.7978, 42.5563, 30.5716, 32.7923,   74.3811, 77.6119, 49.5569, 3.09017, 3.09017, 3.09017, 3.09017,  3.09017, 3.09017, 3.09017, 0.0546329,    0.0546329, -2.5028, -35.0967, -36.2482, -36.2482, -37.5209,     -78.6912, -81.4791}
  spfw = {33.3632, 43.263, 51.6709, 55.5421, 57.1266, 57.2511, 57.3756,    58.059, 58.0778, 58.1995, 56.846, 55.1903, 54.5739, 53.0828,   51.1542, 48.9959, 48.0325, 42.2533, 36.408, 30.7952, 30.1551,  28.6446, 23.138, 19.4168, 6.47053, 5.90328, 
  5.32951, -0.513959, -0.750527, -6.38895, -6.39157, -6.39418,  -6.36456, -6.09357, -6.28599, -5.25369, -4.19539, -2.18625,-0.133803,    2.90414, 6.171}
spfn = {33.3632, 40.2933, 46.5882, 51.9781, 55.5583, 55.5708, 55.5762,   55.4604, 55.4393, 55.0045, 530116, 51.1309, 50.4546, 48.1226,  45.6012, 43.402, 42.066, 37.5522, 32.6864, 28.1979, 28.0685,   25.7067, 17.5943, 13.5547, -2.97428, -3.21054, -3.36422, -5.05466, -5.1301, -6.4392,   -6.76879, -6.48231, -7.20196, -7.00719, -7.53373, -6.00246, -4.41058,    -2.8187, -1.16621, 2.35765, 6.04694}
(*1 STEP*)
myintervals1 = Reap[For[i = 1, i < Length@spfn, i++,
    If[! (((0 < spfn[[i]]) && (0 < spfn[[i + 1]])) || ((spfw[[i]] < 
            0) && (spfw[[i + 1]] < 0))),
     Sow[Interval[{spx[[i]], spx[[i + 1]]}]]]]][[2, 1]]
Print["1)", IntervalUnion @@ myintervals1]

(*2 STEP*)
myintervals2 = Reap[For[i = 1, i < Length@spfn, i++,
    If[spfn[[i]]*spfn[[i + 1]] < 0 && spfw[[i]]*spfw[[i + 1]] < 0,
     Sow[Interval[{Min[xnz[i] = -spmn[[i]]/spkn[[i]], 
         xwz[[i]] = -spmw[[i]]/spkw[[i]]], 
        Max[xnz[[i]] = -spmn[[i]]/spkn[[i]], 
         xwz[[i]] = -spmw[[i]]/spkw[[i]]]}]]
     ]
    ]
   ][[2, 1]]
Print["2)", myintervals2]
POSTED BY: Julia Ilkiv

Hi Julia,

My 2 cents:

IntervalUnion  @@ (Reap[
 For[i = 1, i < Length@spfn, i++, 
  If[! (((0 < spfn[[i]]) && (0 < spfn[[i + 1]])) || ((spfw[[i]] < 
          0) && (spfw[[i + 1]] < 0))), 
   Sow[Interval[{spx[[i]], spx[[i + 1]]}]]]]] // Flatten // Rest)

IntervalUnion only works on a "data type" Interval so to use IntervalUnion you must wrap the list containing the lower and upper bound of an interval, say, {a, b} becomes Interval[{a,b}].

I replaced the Print in your code with an expression that builds a list of such an interval data types by "sowing" it to the upper level of the expression. With Reap the sowed intervals are captured. (The postfix application of Flatten and Rest are just a lazy way to get to the list of sowed intervals, I can recommend to try to evaluate the Reap expression, its result is then self-explaining)

Then you use @@ to apply IntervalUnion to the list of intervals to get your answer:

Interval[{2.11731, 3.3125}, {5.5, 5.75}]

No need for an iterative solution for which you need to maintain a separate state variable.

/Twan
POSTED BY: Twan van der Schoot

Oops, didn't read the first response well enough.
POSTED BY: Twan van der Schoot
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