Abstract:
A visualization and data mining technique can be utilized to facilitate analysis of generated sets of design points for an integrated circuit to enable easy and fast understanding of important properties of generated designs. The use of the visualization and data mining technique significantly reduces the time needed for analysis of design space and decision on which design point to choose for implementing into a circuit design.

Description:
BACKGROUND OF THE INVENTION  
         [0001]    1. Field of the Invention  
           [0002]    The present invention relates to data mining and, more particularly, to a method of visually detecting relationships between values of related goals and variables of a plurality of design points of a circuit.  
           [0003]    2. Description of Related Art  
           [0004]    Heretofore, methods of generating analog circuit designs were based on synthesis programs implemented on a computer. Each synthesis program generates a set of design points for a circuit based upon design variables, design goals and constraints for the circuits. More specifically, each design point is determined from a common set of variables and a common set of goals for the circuit, with each design point having at least one variable or goal different from each other design point.  
           [0005]    Each design point also has a cost or cost value associated therewith. The cost of each design point represents the quality of the design point with respect to a given set of goals, each having a target value and a constraint associated therewith. Design points where the goals better satisfy their design constraints have a lower cost associated therewith.  
           [0006]    Depending on the type of circuit design and the user specified set of information that is input into the synthesis program, the number of design variables and goals vary from several hundred to several thousand. After a synthesis run is complete, a single combination of design variable values and design goal values, i.e., a single design point, is selected. This design point is used as a starting point for an implementation of the circuit on a semiconductor chip. In another application, a set of design points is generated by sampling in the vicinity of a chosen design point. This is typical when the existing “best” design needs to be improved or when it is desired to better understand local variations in the design variables and the design goals.  
           [0007]    Given the large size of the set of design points and the number of variables and goals, it is difficult for a human designer to understand the advantages or disadvantages of selecting a particular design point. It is also difficult to focus on a desired subset of design points by using conventional inspection techniques, such as examining the numerical/textual output of the synthesis program or examining design points one by one. Even when using advanced database querying techniques, significant challenges remain in analyzing this large data set with the large number of design variables and design goals.  
           [0008]    Accordingly, it is desirable to overcome the above problems and others by providing a visual data mining technique that enables analysis of the generated set of design points to provide an easy and fast understanding of important properties of the generated designs; fast, intuitive and easy selection of subsets of design points, or a single design point, with desired properties based on the analysis; and which significantly reduces the time for analysis of the design space and the decision on which design point to choose for practical implementation.  
         SUMMARY OF THE INVENTION  
         [0009]    The invention is a method of selecting one of a plurality of circuit design points to utilize for implementing a circuit. The method includes providing a database having a plurality of design points for a circuit, with each design point determined as a function of a set of goals for the circuit and a set of variables for the circuit. Each design point has a value of at least one goal or one variable that is different from each other design point. A cost is determined for each design point as a function of the set goals for the circuit and, more particularly, as a function of the values of the set of goals for the design point. A cost versus design point chart is displayed having indicia at the intersection of each cost-design point pair. A plurality of indicia is selected in the cost versus design point chart. At least one of the following is displayed: (1) at least one goal variable chart that includes an indicia for the value of the goal and the value of the variable of each design point associated with the selected indicia in the cost versus design point chart; (2) a parallel goal coordinates chart having a plurality of spaced parallel axes each associated with one of the set of goals, with the range of values of each axis related to the range of values of the corresponding goal of the design points associated with the selected indicia in the cost versus design point chart, where the parallel goal coordinates chart includes for each design point associated with the selected indicia in the cost versus design point chart a line that extends through the parallel goal coordinates chart and intersects each axis at the value of a corresponding goal for the design point; and (3) a radar chart having at least three radially extending axes, with each axis representing one of the set of goals, where the radar chart includes a line for each design point associated with the selected indicia in the cost versus design point chart, with each design point line intersecting each axis of the radar chart at the value of the corresponding goal for the design point. One or more of the foregoing charts are utilized to determine which design point to utilize to implement the circuit.  
           [0010]    The method can also include determining at least one of a sensitivity and a correlation for each goal-variable pair associated with each design point associated with the selected indicia in the cost versus design point chart. First and second goal-variable pairs are selected based on the sensitivity or the correlation therefor. First and second goal-variable charts are displayed that includes for each design point of the corresponding first and second goal-variable pairs an indicia related thereto. One of the indicia in the first goal variable chart is selected whereupon a first goal-variable pair of one of the design points is selected. Responsive to the selection of the indicia in the first goal-variable chart, the indicia corresponding to the same design point is highlighted in the second goal-variable chart.  
           [0011]    A constraint can be associated with at least one goal of a design point. Each design point can be classified as feasible where each constraint of the design point is satisfied. Each design point can be classified as infeasible where at least one constraint of the design point is not satisfied.  
           [0012]    The method can also include selecting one goal-variable pair of the set of goals and the set of variables associated with at least the design points associated with a selected indicia. For each design point associated with this selected indicia, the value of the goal and the value of the variable of the selected goal-variable pair can be determined. Variables of a best fit curve equation that represent a best fit line for the thus determined goal and variable values of each design point can be determined. The sensitivity of the selected goal-variable pair can be determined as the function of the best fit line. Also or alternatively, the correlation of the selected goal-variable pair can be determined as a function of the distribution of the goal and variable values of the one goal-variable pair for each design point about the best fit line.  
           [0013]    The invention is also a computer readable medium having stored thereon instructions which, when executed by a processor, cause the processor to display a cost versus design point graph that includes an indicia at the intersection of each cost-design point pair for a circuit. Each design point is determined as a function of a common set of variables and a common set of goals for the circuit where a value of at least one variable or goal of each design point is different from the values of the variables in goals for each other design point. Each cost is determined as a function of the values of the goals for the corresponding design point. A selection of a plurality of indicia in the cost versus design point graph is received and at least one chart is displayed having one of an indicia and a line for each design point associated with the selected indicia in the cost versus design point chart. The at least one chart includes axes for (1) one goal-variable pair; (2) plural goals of the set of goals; or (3) one cost-goal pair. The at least one displayed chart enables selection of one of the design points to utilize for implementing the circuit.  
           [0014]    The instructions can also cause the processor to display a goal versus variable matrix for the design points associated with the selected indicia. The goal versus variable matrix includes in each cell thereof at least one of a sensitivity value and a correlation value for the corresponding goal-variable pair. A selection of first and second goal-variable pairs can be received in the goal-variable matrix. In response to receiving this selection, first and second goal-variable chart can be displayed, with each goal-variable chart including for the goal-variable pair represented thereby an indicia at the intersection of the value of the goal and the value of the variable of each design point associated with the selected indicia in the cost versus design point chart. A selection of one of the indicia in the first goal-variable chart can be received whereupon one of the design points is selected. In response to receiving this selection, the indicia in the second goal-variable chart where the value of the goal and the value of the variable for the same design point is highlighted.  
           [0015]    The instructions can also cause the processor to receive a selection of at least one goal-variable pair of the set of goals and the set of variables associated with the design points associated with the indicia selected in the cost versus design point chart. For the selected goal-variable pair, the value of the goal and the value of the variable for each design point associated with the indicia selected in the cost versus design point chart can be identified. Variables of a best fit curve equation that represent a best fit line for the thus identified goal and variable values of each design point associated with the indicia selected in the cost versus design point chart can be determined for the selected goal-variable pair. The sensitivity of the selected goal-variable pair can then be determined as a function of the best fit line. Also or alternatively, the correlation of the selected goal-variable pair can be determined as a function of the distribution about the best fit line of the goal and variable values of the one goal-variable pair of each design point associated with indicia selected in the cost versus design point graph.  
           [0016]    The chart of plural goals of the set of goals can include a coordinate axis for each of at least three goals of the circuit, with the coordinate axes having a predetermined relation to each other. The graph can also include for each design point associated with the indicia selected in the cost versus design point graph a line that intersects each coordinate axis at a value that corresponds to the value of the corresponding goal for the design point. The graph of plural goals of the set of goals can include a parallel goal coordinates chart having the axes positioned in spaced parallel relation or a radar chart having the axes extending radially from a common center.  
           [0017]    Lastly, the invention is a method of selecting one of a plurality of circuit design points to utilize for implementing a circuit. The method includes providing a database having a plurality of design point for a circuit. Each design point is determined as a function of a common set of goals for the circuit and a common set of variables for the circuit. Each design point has a value of at least one goal or one variable different than each other design point. A set of design points is selected and one of the goals is selected. The values of the selected design points for the selected goal are grouped into at least two groups as a function of the proximity of the values to each other. One of the groups of values is selected whereupon their design points are selected.  
           [0018]    The values of the selected group of values can be grouped into at least two groups as a function of the proximity of the values to each other. Thereafter, the selection of one of the groups of values and the subsequent grouping of the thus selected groups are repeated a desired number of times. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0019]    [0019]FIG. 1 is a chart of design points, variables, goals and costs with associated values in each cell thereof;  
         [0020]    [0020]FIG. 2 is a chart of constraints and target values for the goals of FIG. 1;  
         [0021]    [0021]FIG. 3 is a cost versus design point chart for the costs and design points shown in FIG. 1;  
         [0022]    [0022]FIG. 4 is a sensitivity matrix for each goal-variable pair in FIG. 1;  
         [0023]    [0023]FIG. 5 is a portion of the sensitivity matrix of FIG. 4 and goal-variable charts displayed in response to selecting cells in the sensitivity matrix;  
         [0024]    [0024]FIG. 6 shows a portion of a correlation matrix and a first set of goal-variable charts displayed in response to selecting cells in the correlation matrix;  
         [0025]    [0025]FIG. 7 shows a portion of the correlation matrix and a second set of goal-variable charts displayed in response to selecting cells in the correlation matrix;  
         [0026]    [0026]FIG. 8 is a parallel goal coordinates chart that displays relationships between goals of design points in FIG. 1;  
         [0027]    [0027]FIG. 9 is a goal versus goal chart displayed in response to selecting two goal axes in FIG. 8;  
         [0028]    [0028]FIG. 10 is a cost versus goal chart displayed in response to selecting one goal axis in FIG. 8;  
         [0029]    [0029]FIG. 11 is a radar chart that displays the relationships between three goals and design points shown in FIG. 1; and  
         [0030]    [0030]FIG. 12 shows grouping of goals of selected design points into plural groups as a function of the proximity of the value of the goal for each design point with respect to each other. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0031]    The present invention will be described with reference to the accompanying figures where like reference numbers correspond to like elements.  
         [0032]    The present invention is embodied in a computer software computer program which can be configured to run on a general propose computer or work station of the type known in the art. Generally, the present invention utilizes graphical techniques to visually display data regarding a plurality of design points for a circuit generated by a synthesis program implemented on a computer.  
         [0033]    With reference to FIG. 1, the synthesis program utilizes a common set of variables V 1 , V 2 , V 3 , etc., and a common set of goals G 1 , G 2 , G 3 , etc., for determining design points D 1 , D 2 , D 3 , etc., for a circuit. More specifically, for each design point D, an appropriate value is associated with each variable V and each goal G wherefrom the synthesis program determines the performance characteristics of the circuit for these variables V and goals G. The combination of the values of the variables V and the goals G along with the corresponding performance characteristics of the circuit represent the design point for the values of the variables V and the goals G. Stated differently, each design point is a function of the values of the variables V and the goals G utilized to determine said design point.  
         [0034]    Each design point also has a cost C1, C2, C3, etc., associated therewith. A general form algorithm for determining the cost of each design point is shown in the following equation (EQ1). However, it is to be appreciated that other suitable algorithms can be utilized to determine the cost of each design point.  
       Cost   =       ∑     i   =   1     n                       f        (     G   i     )            (     w   i     )                               
 
         [0035]    where i=design point number;  
         [0036]    n=total number of design points;  
         [0037]    G i  are goals related to the cost;  
         [0038]    w i  are weights used to weigh the contribution of each goal; and  
         [0039]    f(G i ) is an objective function that is utilized to determine the extent to which the corresponding goal is satisfied.  
         [0040]    With reference to FIG. 2, and with continuing reference to FIG. 1, each goal G has associated therewith a design constraint that places a restriction or requirement on a target value for the goal. In FIG. 2, the column “Target Value” includes comparison values and the column “Constraint” contains comparison operators. The comparison operator “minimize” (or “maximize”) means that the value of the corresponding goal should be minimized (or “maximized”) and the target value for that goal is given only as a suggestion to the synthesis program. Constraints can have one of the following operators: minimize, maximize, less than and greater than.  
         [0041]    With reference to FIG. 3, and with continuing reference to FIG. 1, a plurality of design points generated by the synthesis program are stored in a design database. At an appropriate time, a designer can launch a cost versus design point chart  2  that includes indicia  4  at the intersection of each cost and its design point. Indicia  4  having a first appearance or color, e.g., red, represents infeasible design points while indicia  4  having a second appearance or color, e.g., green, represents feasible design points. A feasible design point is one where the constraints associated with the goals of the design point can be met. In contrast, an infeasible design point is one where at least one constraint of one goal cannot be met.  
         [0042]    Next, utilizing appropriate techniques, a designer selects a region  6  of design points to explore. Once region  6  is selected, the designer can perform a sensitivity analysis, a correlation analysis, a goal trade-off analysis, a corners analysis and/or a cluster analysis. The use of all or part of these analyses and the order in which they are utilized are not fixed.  
         [0043]    If, as a result of one or more of the foregoing analyses, the “best” design point is found, the design process is finished. Otherwise, the designer can make adjustments to the values of the variables and/or the goals based upon observations about the current design space, and the synthesis program can be run again with these adjustments. This other synthesis run can be initiated from the beginning of the design evaluation process, or may generate design points in addition to the existing design points. This process is repeated until the “best” design point is found.  
         [0044]    Cost versus design point chart  2  includes a sensitivity button  8 , a correlation button  10 , a goal trade-off button  12 , a comers button  14  and a cluster button  16  which the designer can select in any order to display a corresponding chart, graph or matrix that can be utilized by the designer for elevating design points. The response of the computer software to the activation of each of these buttons  8  to  16  will now be described.  
         [0045]    With reference to FIG. 4, and with continuing reference to FIGS. 1 and 3, in response to selecting sensitivity button  8 , a sensitivity matrix  20  is displayed. Sensitivity matrix  20  includes a matrix of sensitivity cells  22  for the values of the goal G and variable V for each goal-variable pair utilized to determine each design point D in FIG. 1. Each sensitivity cell  22  includes a value that represents a sensitivity of the corresponding goal G to the corresponding variable V, or vice versa. Sensitivity matrix  20  can also include an “Average” column  24  having average cells  26  therein where the value included in each average cell  26  is the average of the values of the goals G for the corresponding variable V. For example, the value 0.01 included in the cell at the intersection of Variable V 1  and “Average” column  24  is the average of the values included in sensitivity cells  22  for each goal G associated with Variable V 1 .  
         [0046]    The sensitivity value included in each sensitivity cell  22 , i.e., each cell at the intersection of a variable V and a goal G, can be determined utilizing the following equation (EQ2) or any other suitable equation:  
         EQ                 2        :                                       S   =       S   xy     /     S   xx                     where                   S   xy       =         ∑     i   =   1     n                       x   i          y   i         -       (     Σ                   x   i       )            (     Σ                   y   i       )     /   n           ;                            S   xx     =         ∑     i   =   1     n                     x   i   2       -         (     Σ                   x   i       )     2     /   n         ;                           
 
         [0047]    S=sensitivity;  
         [0048]    i=design point number;  
         [0049]    n=total number of design points;  
         [0050]    x i =variable value of design point i; and  
         [0051]    y i =goal value of design point i.  
         [0052]    The sensitivity value for each goal-variable pair is determined utilizing equation EQ2 for the goal and variable values for each design point D shown in FIG. 1. For example, for the goal-variable pair G 1 -V 1 , the value of goal G 1  and variable V 1  values for each design point D are utilized in equation EQ2 to determine the sensitivity for goal variable pair G 1 -V 1 .  
         [0053]    The value included in each sensitivity cell  22  represents the extent to which the goal changes when the variable changes its value for a unit interval. The value of sensitivity S for each goal-variable pair is a number between ±1.0. A high sensitivity value, i.e., one that is closer to +1 or −1, means that the value of the goal changes significantly for small changes in the value of the variable. In contrast, a low sensitivity value, i.e., one closer to 0, means that changes of the value of the goal are small or negligible for changes in the value of the variable.  
         [0054]    Each cell  22 ,  26  can be color coded depending on its sensitivity value such that when the absolute sensitivity value is close to ±1.0, the cell color is brighter. This color coding enables easy identification of entries with higher sensitivity values in the matrix.  
         [0055]    Utilizing suitable techniques, each row or column can be individually selected to display sensitivity values in increasing or decreasing order. For example, the designer can sort on average sensitivity values to obtain variables V that have the strongest average impact on all goals G. Moreover, utilizing appropriate techniques, columns of sensitivity matrix  20  can be moved freely so that goals G in which the designer is most interested can be brought close together and their sensitivities analyzed.  
         [0056]    With reference to FIG. 5, and with continuing reference to FIGS. 1 and 4, utilizing appropriate techniques, one or more sensitivity cells  22  can be selected. For each sensitivity cell  22  selected, a corresponding goal-variable chart is displayed. For example, in response to selecting the sensitivity cell  22  at the intersection of goal-variable pair G 23 -V 18  in FIG. 5, a goal-variable chart  30  is displayed. Similarly, in response to selecting the sensitivity cell  22  at the intersection of goal-variable pair G 22 -V 19 , a goal-variable chart  32  is displayed. Similarly, selecting any other sensitivity cell  22  causes a corresponding goal-variable chart to be displayed.  
         [0057]    Each goal-variable chart  30  and  32  includes a plurality of indicia  34 , each of which represents a design point D in FIG. 1. More specifically, the indicia  34  for each design point D is located in each goal-variable chart  30  and  32  at the intersection of the value for the goal and the value of the variable for the design point D in the chart. For example, suppose that indicia  34 - 1  in goal-variable chart  30  relates design to point D 1 . The position of indicia  34 - 1  in goal-variable chart  30  is based upon the value of goal G 23  and the value of variable V 18  for design point D 1 . Similar comments apply in respect of all other indicia  34  in goal-variable charts  30  and  32 .  
         [0058]    The slopes of imaginary lines  36  and  38  formed by indicia  34  in goal-variable charts  30  and  32 , respectively, are higher for higher sensitivity values and lower for lower sensitivity values. A positive sloping line  36  or  38  indicates a direct relationship between the goal-variable pair and a negative sloping line  30  or  32  indicates an inverse relationship between the goal-variable pair. By evaluating goal-variable charts  30  and  32 , a designer can access the sensitivity of each goal-variable pair that has been selected in the sensitivity matrix.  
         [0059]    With reference to FIGS. 6 and 7, and with continuing reference to FIGS. 3 and 1, in response to the selection of correlation button  10  in cost versus design point chart  2 , a correlation matrix  40  is displayed. Correlation matrix  40  includes a correlation cell  42  for each goal-variable pair of the goals G and variables V utilized to determine each design point D in FIG. 1. Correlation matrix  40  also includes an “Average” column  44  in which the average cells  46  are arranged. Each average cell  46  includes a value that is the average of the correlation values of the goals G for the corresponding variables V. The correlation for each selected goal-variable pair can be determined utilizing the following equation (EQ3) or any other suitable equation:  
         EQ                 3        :                                       r   =       S   xy     /       (       S   xx          S   yy       )       1   /   2                               where                   S   xy       =         ∑     i   =   1     n                       x   i          y   i         -       (     Σ                   x   i       )            (     Σ                   y   i       )     /   n           ;          
                         S   xx     =         ∑     i   =   1     n                     x   i   2       -         (     Σ                   x   i       )     2     /   n         ;          
                         S   yy     =         ∑     i   =   1     n                     y   i   2       -         (     Σ                   y   i       )     2     /   n         ;               □                             
 
         [0060]    r=correlation;  
         [0061]    i=design point number;  
         [0062]    n=total number of design points;  
         [0063]    x i =variable value of design point i; and  
         [0064]    y i =goal value of design point i.  
         [0065]    The correlation value for each goal-variable pair is determined utilizing equation EQ3 for the goal and variable values for each design point D shown in FIG. 1. For example, for the goal-variable pair G 2 -V 2 , the goal G 2  and variable V 2  values for each design point D are utilized in equation EQ3 to determine the correlation r for the goal-variable pair G 2 -V 2 .  
         [0066]    As shown in FIG. 6, in response to selecting correlation cell  42  at the intersection of a goal-variable pair, a corresponding goal-variable chart having indicia related to each design point is displayed. For example, in response to selecting correlation cells  42  at the intersections of goal-variable pairs G 2 -V 2  and G 3 -V 1 , goal-variable charts  48  and  50  are displayed having indicia  52  corresponding to each design point D in FIG. 1. For example, suppose that indicia  52 - 1  in goal-variable chart  48  corresponds to design point D 1 . In this case, the location of indicia  52 - 1  in goal-variable chart  48  is based on value of variable V 2  for design point D 1  and the value of goal G 2  for design point D 1 . Similar comments apply in respect of the position of the other indicia  52  in each goal-variable chart  48  and  50 . In FIG. 6, groups of indicia  52  are shown aligned in columns. This can occur when the values of variables V 2  and V 1  have discrete values. However, this is not to be construed as limiting the invention since the values of variables V 1  or V 2  can vary continuously whereupon indicia  52  in charts  48  or  50  can be arranged in a scatterplot arrangement like indicia  34  shown in FIG. 5.  
         [0067]    As can be seen, sensitivity matrix  20  and correlation matrix  40  are organized in the same manner. The only difference is that the values displayed in correlation matrix  40  represent correlation values for each goal-variable pair, and their range is between 0.0 and 1.0, with 0.0 representing a low correlation and with 1.0 representing a high correlation value. Each correlation value indicates the degree of linearity between the value of the goal and the value of the variable for the corresponding goal-variable pair.  
         [0068]    A designer can evaluate goal-variable charts  48  and  50  to determine the linearity of the relationship between the corresponding goal-variable pair. This information aids the designer in selecting a suitable design point for implementation.  
         [0069]    Since both similarity and correlation information are relevant, sensitivity matrix  20  and/or goal-variable charts  30  and  32  can alternatingly be displayed with correlation matrix  40  and/or goal-variable charts  48  and  50  in a manner known in the art. Alternatively, correlation values and sensitivity values can be displayed together in the same cell of a combination sensitivity/correlation matrix (not shown). Selecting one of the cells of this sensitivity/correlation matrix can cause a goal-variable chart for sensitivity values and/or a goal-variable chart for correlations to be displayed for use by the designer.  
         [0070]    With reference to FIG. 7, utilizing goal-variable charts of the type shown in FIGS.  5  or  6 , a designer can utilize the present invention to gain an understanding about goal and variable changes when a different design point is selected. For example, the designer can select a plurality of correlation cells  42  for goals of a single variable. For example, the designer can select goals G 1 , G 2 , G 3  and G 4  for variable V 1 . In response to selecting these correlation cells  42 , goal-variable charts  60 ,  62 ,  64  and  66  are displayed. Similar comments apply in respect of the selection of a plurality of sensitivity cells  22  for goals of a single variable.  
         [0071]    Corresponding indicia in charts  60  to  66  can be linked. Hence, in response to selection of an indicia  68 - 1  in chart  60 , indicia  68 - 2 ,  68 - 3  and  68 - 4  for the same design point are highlighted in charts  62  to  66 . Similar comments apply in respect of selecting indicia  34 ;  52 ; or  68  in any of charts  30  and  32 ;  48  and  50 ; and  60  to  66  and highlighting indicia for the corresponding design point in one of the other charts  30  and  32 ;  48  and  50 ; and  60  to  66 , respectively. The size of each indicia  34 ,  52  and  68  can be related to the cost of the design point represented thereby. For example, the size of the indicia can be inversely proportional to the cost. Each chart  30 ,  32 ,  48 ,  50  and  60  to  66  can include an indicator  72  that points in the direction of feasible goal values. The indicator  72  in each chart can be color coded so that a first color represents a strict constraint (greater than or less than) and a second color represents a preferential constraint (maximize or minimize).  
         [0072]    The designer can also highlight in one of charts  60  to  66  an indicia  70  that represents an end point for the analysis whereupon corresponding indicia for the same design point are highlighted in the other charts. The highlighting of indicia  70  in each chart  60  to  66  can be utilized to avoid the designer from analyzing undesirable design points.  
         [0073]    By observing changes in goal and variable values in this way for several design points, the designer can gain an understanding of changes in the design space and select a design point that best satisfies the design specification. From the example shown in FIG. 7, it can be seen that the same change in a variable value can cause very different changes in different goal values. Also, the change in goal values does not always follow a general relationship, e.g., linear relationship, suggested by observing all points in charts  60  to  66 .  
         [0074]    With reference to FIG. 8, and with continuing reference to FIG. 3, in response to selecting goal-trade off button  12 , a parallel goal coordinates chart displays multiple goal values on parallel axes. Each line  82  in chart  80  represents a single design point. Each line  80  is color coded to indicate the cost associated therewith. The key to the color code is given in a color bar  84 . The range of cost values displayed on chart  80  can be selected by moving a cost range slider  86  adjacent color bar  84  upwardly or downwardly until the desired range of cost values is displayed. A slider  88  enables adjustment of which goals are displayed in chart  80 . Desirably, all goals are displayed. The range of each goal axis in FIG. 8 is determined from the data currently displayed. Hence, each goal axis can have a different data range.  
         [0075]    Utilizing parallel goal coordinates chart  80 , a designer can observe general trends in data with respect to their cost and goal values. Lines  82  having a first color, e.g., green, (indicated by a solid line) indicate lower cost. By following each line  82  having the first color, a designer can observe changes of individual goals for more acceptable design points. Similarly, by following each line having a second or a third color, e.g., yellow (indicated by a dash-dot-dash line) or red (indicated by a dash-dot-dot-dash line), the designer can observe goal trends and changes for less desirable design points.  
         [0076]    The designer can also observe relationships between adjacent goals in the chart and determine whether direct or inverse relationships between goals exist by observing whether lines  82  are parallel (direct relationship) or crossed (inverse relationship). Each goal axis in parallel goal coordinates chart  80  can be dragged to an arbitrary position within the chart utilizing suitable techniques. This facilitates comparison of any two goals even if the goal axes were not initially adjacent.  
         [0077]    As discussed above, moving cost-range slider  86  upwardly or downwardly applies visual filtering to the data displayed in parallel goal coordinates chart  80 . For example, moving slider  90  upwardly toward slider  92  decreases the range of costs displayed in chart  80  by increasing the value of the lowest cost displayed. Conversely, moving slider  92  toward slider  90  decreases the range of costs displayed on chart  80  by decreasing the value of the greatest cost displayed. Still further, sliders  90  and  92  can be moved toward each other so that the range of cost displayed on chart  80  decreases as a result of decreasing the greatest cost displayed in response to moving slider  92  downward and by increasing the least cost displayed in response to moving slider  90  upward.  
         [0078]    Using cost-range slider  86 , the number of design points displayed on chart  80  can be reduced significantly thereby making analysis of individual lines  82  (design points) easier. To this end, utilizing suitable techniques, a single line can be selected whereupon the goal closest adjacent the point where the line was selected and the cost for the selected design point can be displayed (not shown).  
         [0079]    With reference to FIG. 9, and with continuing reference to FIGS. 3 and 8, the present invention is configured so that selecting two different goals in parallel goal coordinates chart  80  displays a goal versus goal chart  100 . While goal versus goal chart  100  shows goal G 1  versus G 2 , this is not to be construed as limiting the invention since the two selected goals do not need to be adjacent. Goal versus goal chart  100  includes indicia  102  for each design point D of FIG. 1. More specifically, each indicia  102  is positioned at the intersection of the value of goal G 1  and the value of goal G 2  for the corresponding design point D. Goal versus goal chart  100  can also include goal target lines  104  and  106  located on chart  100  at the target value for the corresponding goal. For example, as shown in FIG. 2, goal G 1  has a target value of 0.1. Hence, in FIG. 9, target line  104  intersects the goal G 1  axis at a value of 0.1. Similarly, in FIG. 2, goal G 2  has a target value of 0.02. Hence, in FIG. 9, target line  106  intersects the goal G 2  axis at a value of 0.02.  
         [0080]    Each target line  104  and  106  can also be color coded in accordance with the type of constraint on the corresponding target value. For example, if a target line represents a strict constraint, such as greater than or less than, the line can be a first color, e.g., blue, whereas if a constraint is a preferential constraint, such as minimize or maximize, the line can be a second color, e.g., red.  
         [0081]    The intersection of each target line  104  and  106  with its axis can include a suitable indicator  110  and  112 , respectively, which can point toward goals having feasible values. The size of each indicia  102  can be a function of the cost associated with the corresponding design point. For example, indicia  102  having a large size can indicate a low cost and vice versa. The color of each indicia  102  can also indicate whether the corresponding design point is feasible or infeasible.  
         [0082]    Utilizing goal versus goal chart  100 , the designer can observe a feasibility region  108  for the two selected goals. To this end, if one of the two selected goals is the most important, then selection of a suitable indicia  102  in feasibility region  108  can be made. However, if other goals need to be considered, the designer can analyze relationships for those goals for different goal versus goal pairs in a manner shown in FIG. 9. Moreover, in a manner similar to a plurality of charts  60  to  66  in FIG. 7, a plurality of goal versus goal charts of the type shown in FIG. 9 can be concurrently displayed with indicia for each design point in each chart linked to indicia for the corresponding design point in the other charts. In this manner, the designer can analyze how changes in one pair of goals influence changes in another pair of goals.  
         [0083]    With reference to FIG. 10, and with continuing reference to FIG. 8, in response to selecting a single axis of parallel goal coordinates chart  80  twice in succession, a cost versus goal chart  120  is displayed. For example, if the goal G 1  axis in FIG. 8 is selected twice in succession, cost versus goal chart  120  is displayed.  
         [0084]    Cost versus goal chart  120  includes indicia  122  positioned at the intersection of the value of the cost and the value of the goal for each design point D in FIG. 1. Indicia  122  representing feasible design points can have a first, e.g., green, color while indicia  122  representing infeasible design points have a second, e.g., red, color. The size of each indicia  122  in chart  120  can be the same or feasible design points can have a different size than infeasible design points. As can be seen in cost versus goal chart  120 , the cost value C decreases as the value of goal G 1  increases. The same effect can also be observed in FIG. 9 if the size of each indicia  102  is related to the cost associated with the corresponding design point.  
         [0085]    Other goals can also influence the cost function in an opposite direction from goal G 1 . To gain additional insight into influences of other goals to the cost function, the designer can use multiple cost versus goal charts with linked views and select different indicia on different charts to observe changes across different coordinates in a manner similar to that discussed above in connection with charts  60  to  66  in FIG. 7. Alternatively, the designer can return to visual filtering by cost utilizing cost range slider  84  of parallel goal coordinates chart  80 .  
         [0086]    With reference to FIG. 11, and with continuing reference to FIG. 3, in response to selecting corners button  14 , a goal selection menu  130  and a display button  132  are displayed. In response to selecting three goals, e.g., goal G 1 , G 2 , and G 3 , in goal selection menu  130  and thereafter activating display button  132 , a radar chart  134  is displayed where each goal is represented by an axis that extends radially from a common center. It is desirable in radar chart  134  that each axis extends in a different direction to facilitate discrimination among lines  136  representing design points D in FIG. 1. In the example shown in FIG. 11, the three axes are spaced 120° relative to each other. However, this is not to be construed as limiting the invention since the number of goals selected in goal selection menu  130  will determine the number of axes and their angular spacing with respect to each other around the common center.  
         [0087]    Each line  136  can be color coded based on its cost. For example, a first color, e.g., green, can be utilized to indicate a more favorable cost, a second color, e.g., red, can be utilized to indicate a less favorable cost and a third color, e.g., yellow, can be utilized to indicate an intermediate cost.  
         [0088]    A color bar  140  and a cost-range slider  142  can be provided and used in the manner described above for color bar  84  and cost range slider  86  in FIG. 8 for filtering cost and goals based on color.  
         [0089]    Because of imperfect manufacturing processes and real world environmental conditions, a designer typically validates a design across several different corners. For example, if a design is expected to work at temperatures between 0° C. and 100° C., the designer will simulate the design at least at the two extremes or corners, 0° C. and 100° C. Hence, in radar chart  134 , one goal axis will be for 0° C. and another goal axis will be for 100° C. The third axis can then be another goal such, as the minimum operating voltage.  
         [0090]    Axes of radar chart  134  can also be specified using more than one manufacturing or environmental condition. For example, if a circuit must work between 0° C. and 100° C. and between 2 volts and 2.5 volts, radar chart  134  will have four corners, namely, 0° C. at 2 volts; 0° C. at 2.5 volts; 100° at 2 volts; and 100° C. at 2.5 volts.  
         [0091]    Because of the exponential growth in the number of corners when considering different manufacturing and environmental conditions, typically, between one and ten corners are chosen. These corners are representative corners. However, this number of corners is not to be construed as limiting the invention. Once the goal meets these corners, there is a good chance that the goal will meet all corners. Once an initial design process is complete, the design may be further validated across all or a larger set of corners to ensure that the circuit works correctly.  
         [0092]    It has been observed that it is often difficult to meet a goal value for a particular corner. This can occur for several reasons. One reason may be that the design is overly sensitive at a particular corner. Another may be that redesigning the circuit to meet a goal at one corner can hurt the performance in another corner thus causing a design, redesign, “ping pong” effect that wastes time. Radar charts of the type shown in FIG. 11 can help provide design insight into how a goal is changing from one corner to the next. Specifically, a designer focusing on one corner of radar chart  134  can observe the relationship in the other corners of the radar chart. This enables the designer to observe, among other things, whether the goal at a particular corner is overly sensitive or if there is an inverse relationship between two corners. An inverse relationship implies that there are two competing comers. This could signify a “ping pong” condition. Another important observation that can be made from radar chart  134  is the relative values for a goal at each corner. This enables assessment of how much a goal value changes when moving from one corner to another.  
         [0093]    With reference to FIG. 12, and with continuing reference to FIG. 3, in response to selecting cluster button  16 , a goal selection menu  150  is displayed. Goal selection menu  150  is configured so that only one goal at a time can be selected. Once a goal in goal selection menu  150  has been selected, the values of the selected goal for all of the design points D are clustered into two or more groups  154  as a function of their values and these groups  154  are displayed. For example, as shown in FIG. 12, in response to selecting goal G 5  in goal selection menu  150 , the values of goal G 5  for ten different design points are grouped together, as shown, into Groups I, II and III. However, the number of groups can be changed if desired. The number of groups can be determined automatically or manually. In the example shown in FIG. 12, values of the selected goal are grouped into three separate groups or clusters. The labels “Group I”, “Group II” and “Group III” are assigned to each cluster according to the values of the goal.  
         [0094]    At this point, selection can be made of which group best suits the requirements of the circuit design. For example, for operational amplifiers, it is common to choose a high gain group because the goal is to have gain as high as possible. Hence, if goal G 5  in FIG. 12 is the gain goal, Group III is selected. This selection reduces the number of design points for further exploration.  
         [0095]    Next, the same or another goal can be selected in goal selection menu  150 . If the same goal, i.e., goal G 5 , is selected, the values of the selected goal for all of the design points D associated with selected group  154  are clustered into two or more groups (not shown) as a function of their values and these groups are displayed. However, if a different goal, e.g., goal G 7 , is selected, the values of the selected goal for all of the design points D associated with selected group  154  are clustered into two or more groups  156  as a function of the values of goal G 7  for these design points D and these groups  156  are displayed. The sequential selection of the same or different goals in goal selection menu  150  forms the clustering tree  158  shown in FIG. 12. The process of selecting one of the displayed groups at each level of incremental clustering tree  158  continues until a desired number of design points D have been isolated.  
         [0096]    Concurrent with the selection of each new goal in goal selection menu  150 , a parallel goal coordinates chart, of the type shown in FIG. 8, which includes for each goal selected in goal selection menu  150  a parallel axis and a line for each design point D of the selected group in the lowest level of incremental clustering tree  158 , can be displayed.  
         [0097]    A serialize button  160  can be selected for saving the search strategy displayed in incremental clustering tree  158  to a file for subsequent retrieval and analysis.  
         [0098]    As an alternative to producing incremental clustering tree  158  in the above described manner, a simple script file (not shown) can be prepared that includes the order of importance of analyzed goals and the desired range of values for each goal. In response to activating cluster button  16  in FIG. 3, the script file can be executed to produce an incremental clustering tree, like incremental clustering tree  158 , but without having to incrementally select goals in the manner described above.  
         [0099]    The present invention has been described with reference to the preferred embodiment. Obvious modifications and alterations will occur to others upon reading and understanding the preceding detailed description. It is intended that the invention be construed as including all such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof.