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2 2, 3 3, and 4 4. These may be instantiated with the proper constructors, passing a list of column vectors, the proper total number of elements in column-major order, or with a single oat that lls in the diagonal, leaving all other elements 0.

because d2 1.128 when two consecutive observations are used to calculate a moving range. It is also possible to establish a control chart on the moving range using D3 and D4 for n 2. The parameters for these charts are de ned as follows.

//define the 2x2 matrix: // | 1.0 2.0 | // | 3.0 4.0 | float2x2 m = float2x2(1.0, 3.0, 2.0, 4.0); float2 col0 = float2(1.0, 3.0); float2 col1 = float2(2.0, 4.0); float2x2 m = float2x2(col0, col1);

The center line and upper and lower control limits for a control chart for individuals are UCL CL LCL x x x mr 3 d2 x mr 3 1.128 3 mr d2 x 3 mr 1.128 (16-19)

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The procedure is illustrated in the following example. EXAMPLE 16-2 Table 16-3 shows 20 observations on concentration for the output of a chemical process. The observations are taken at one-hour intervals. If several observations are taken at the same time, the observed concentration reading will differ only because of measurement error. Since the measurement error is small, only one observation is taken each hour. To set up the control chart for individuals, note that the sample average of the 20 concentration readings is x 99.1 and that the average of the moving ranges of two observations shown in the last column of Table 16-3 is mr 2.59. To set up the moving-range chart, we note that D3 0 and D4 3.267 for n 2. Therefore, the moving-range chart has center line mr 2.59, LCL 0, and UCL D4mr 13.267212.592 8.46. The control chart is shown as the lower control chart in Fig. 16-11 on page 618. This control chart was constructed by Minitab. Because no points exceed the upper control limit, we may now set up the control chart for individual concentration measurements. If a moving range of n 2 observations is used,

You can nd an element with double bracket notation, column then row, or pull out a column vector with a single pair of brackets.

The control chart for individual concentration measurements is shown as the upper control chart in Fig. 16-11. There is no indication of an out-of-control condition. The chart for individuals can be interpreted much like an ordinary X control chart. A shift in the process average will result in either a point (or points) outside the control limits, or a pattern consisting of a run on one side of the center line. Some care should be exercised in interpreting patterns on the moving-range chart. The moving ranges are correlated, and this correlation may often induce a pattern of runs or cycles on the chart. The individual measurements are assumed to be uncorrelated, however, and any apparent pattern on the individuals control chart should be carefully investigated. The control chart for individuals is very insensitive to small shifts in the process mean. For example, if the size of the shift in the mean is one standard deviation, the average number of points to detect this shift is 43.9. This result is shown later in the chapter. While the performance of the control chart for individuals is much better for large shifts, in many situations the shift of interest is not large and more rapid shift detection is desirable. In these cases, we recommend the cumulative sum control chart (discussed in Section 16-10) or an exponentially weighted moving-average chart (Montgomery, 2001).

//pull out element 2.0 float element = m[1][0]; //pull out column vector <1.0, 3.0> float2 firstCol = m[0];

Table 16-3 Chemical Process Concentration Measurements Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 x Concentration x 102.0 94.8 98.3 98.4 102.0 98.5 99.0 97.7 100.0 98.1 101.3 98.7 101.1 98.4 97.0 96.7 100.3 101.4 97.2 101.0 99.1 mr Moving Range mr 7.2 3.5 0.1 3.6 3.5 0.5 1.3 2.3 1.9 3.2 2.6 2.4 2.7 1.4 0.3 3.6 1.1 4.2 3.8 2.59

The arithmetic operators are overloaded for matrices, too. In this case, multiplication of matrices with the same dimension actually performs matrix multiplication, not per-element multiplication. Multiplication of vectors and matrices with the proper dimension in the proper order also performs matrix multiplication. For example,

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